Reptile Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⢿⣷⣿⣾⣷⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢤⡟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⡿⠋⠉⠉⠉⠛⢿⣿⡿⣦⠀⠀⠀⠀⠀⠀⢀⣾⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣮⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣻⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠀⢐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡉⣹⢦⣀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠈⠛⢿⣿⣍⡳⣦⣀⠀⠀⠀⠻⣿⡿⣿⣶⣿⣿⠟
⠀⠀⠀⠀⠀⣴⣻⠟⠛⠛⠿⣿⣷⣄⠀⠀⠀⠈⠛⠷⣧⣿⡓⣦⡀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⢨⢿⡏⠀⠀⠀⠀⠘⢿⢻⣆⠀⠀⠀⠀⠀⠈⠙⢷⣿⣟⣦
⠀⠀⠀⠀⠀⢾⣳⣀⠀⠀⠀⠀⠘⣿⢿⡆⠀⠀⠀⠀⠀⠀⠀⢻⡿⢽⣇
⠀⠀⠀⠀⠀⠈⠙⢽⣶⣤⡀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣷⡞⠛
⠀⠀⠀⠀⢤⣄⠀⠀⠉⠻⢿⣶⡄⠀⠹⡿⣷⡄⠀⠀⠀⠀⣠⣾⣿⣯⠅
⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠹⣿⡄⠀⠹⣟⣛⢶⢶⣶⢾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⢀⣿⠁⠀⠀⠀⠙⠚⠮⠼⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣦⣰⣲⣆⡒⠒⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣹⡝⡋⠭⠰⠦⠰⠄⡌⢀⡃⠀⢮⣙⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡠⠺⣏⣷⣎⣤⣉⣀⠀⠀⠀⠀⠐⠈⠅⢙⠂⣈⢢⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣒⣲⠭⠓⢀⡼⣙⣧⡽⣯⡞⠋⠀⡠⠙⢢⠀⠀⠀⠀⠀⠀⠙⢄⡜⣎⣹⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡖⢻⠷⠚⠉⣠⡤⢒⡏⣾⡉⣿⡗⢋⡀⠀⠀⣿⡞⣦⣕⣤⣀⡀⠀⢀⡀⢆⡤⠛⡭⡇
⠀⠀⠀⠀⠀⢠⣾⣍⡗⠁⣰⡶⠋⠁⢀⡾⡰⠿⣿⢯⠻⢷⣨⠃⠀⠰⡻⣧⣼⣿⠒⣿⣽⢙⣳⣆⣾⠟⣽⠃
⠀⠀⠀⠀⢠⡿⣥⡏⢀⡞⠁⠀⠀⠀⣾⢿⢿⣷⣿⡏⠉⠉⠀⠀⠀⠀⠙⠦⣛⣩⡛⡿⣛⣻⡿⣫⣷⠟⠁
⠀⠀⠀⠀⣼⡞⢻⡄⠸⣇⣀⣀⣠⢞⣷⣷⣴⢯⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉
⠀⠀⠀⠀⢹⡛⢿⢛⣷⣶⠞⢴⣾⣏⣹⠿⢛⡜⠁
⠀⠀⠀⠀⠀⠳⣬⣟⠻⠿⠷⡿⠛⠛⣱⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠿⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣤⣤⣶⡿⢿⣻⣿⣿⣿⣿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣀⣴⣴⣶⢿⡾⣷⡟⠙⡿⠿⣿⡇⡙⣿⣿⣿⣯⣿⣿⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡞⢛⣿⣯⣽⣿⣿⣿⠥⢄⣀⡙⠙⠠⢰⠶⣾⣷⣿⢉⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⡴⣋⣭⢿⣿⣿⣿⠿⠿⠾⠿⠿⣿⣿⣷⣶⢉⣦⣦⡼⠟⠋⣽⣿⡶⢿
⠀⠀⠀⠀⢀⠔⣽⣿⡿⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧⠌⠉⠀⠀⠀⠀⠘⠿⢧⣤⡄
⠀⠀⠀⢀⣾⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢿⡧⠄
⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠋
⠀⠀⠀⠸⢹⡇
⠀⠀⠀⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡶⣦⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⠾⠻⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⠀⠀⠀⠀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣘⣿⢯⣴⣄⣶⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⣿⣧⣟⡋⣿⡆⣇⡿⠏⠛⠿⠛⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣏⡿⣽⣿⣿⣿⣽⡟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡛⣿⣿⣿⣷⡾⠟⠈⢹⢾⣻⡤⣤⡾⣦⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣝⡃⣿⠻⠋⠁⠀⠀⠀⠈⠓⠋⠉⠁⠉⡍⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣦⣒⡶⢹⢞⣿⣿⣷⣒⡶⡄
⠀⠀⠀⣠⡴⠶⠚⠒⢦⡀⠀⠀⢀⠀⡟⣿⠛⠻⣶⣻⣷⡶⠛⠛⣿⣥⠇
⠀⢠⡾⠋⠀⠀⢀⣀⣸⠇⢠⢭⣞⣶⣿⡟⠀⢰⠛⢛⡟⠀⠀⢈⣷⣅⣀⡠⠄
⢠⣿⠃⠀⠀⠀⠀⠉⠀⠀⠀⠢⠍⠁⠁⠀⢠⡟⢷⡿⠁⠀⠀⠀⠛⠻⢿⡿⠬⠁
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢷⡞⠁⠀⠀⠀⠀⠀⠀⠐⠈⠃
⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⠏
⠀⠘⣿⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⠁
⠀⠀⠈⠛⢿⣿⣶⣲⣖⣾⣹⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉


⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⣀⣠⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⢀⣤⣶⣾⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣰⣿⣿⠿⠛⠛⢿⣿⣿⣶⣶⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣀⠀⠀⣿⣿⣇⣠⣤⣶⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⠀⠀⠀⠀⠀⠙⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢻⣿⣿⠉⠁⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⣄⠀⠀⢿⣿⣧⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⢀⣰⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠉⠈⠉⠻⣷⡄⠈⢻⣿⣇⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⢸⣷⠀⢸⣿⣿⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⣸⡟⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣷⣦⣤⣾⣯⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⡴⠞⠉⠁⠈⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⢤⠂⠀⠀⠀⠠⠤⢤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⡊⠀⠨⠀⠀⠀⡶⠲⠓⡧⠄⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢩⠈⠂⢢⡼⠒⠂⢴⠨⠛⠚⡓⠀⠀⠠⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⡈⢆⢢⠜⠁⠀⠀⠀⠀⠈⠚⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⠅⠢⠌⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠐⢈⣉⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠠⠤⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠠⠐⠂⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⠂⠬⠐⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⠐⠂⠩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⣁⠐⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠂⠠⠈⠩⡄
⠀⠀⠀⠀⠀⣀⢴⠶⣺⡉⢭⣣⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠡⠐⣀⠠⢇
⠀⠀⠀⢀⣼⡋⡐⢉⡄⠋⠁⠀⠀⠈⠘⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠂⢅⠐⢺
⠀⠀⢀⡖⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠔⠲⣈⡛
⠀⠀⢸⣍⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⡇⠀⠀⠀⠀⠀⠀⠀⡟⡐⢊⡑⢈⡇
⠀⠀⢸⡄⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⣰⠢⠌⠤⠑⣼
⠀⠀⠘⠦⢈⡁⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢍⠡⡉⢄⡱⠃
⠀⠀⠀⠱⡂⠠⠀⠜⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡚⠋⠠⣈⠂⠠⡜⠁
⠀⠀⠀⠀⠙⢀⠒⢈⠥⢋⠗⠶⢤⡄⢠⣤⣤⡤⠤⡒⢲⡉⠇⡐⠨⣀⡂⠠⣨⠎
⠀⠀⠀⠀⠀⠀⠰⢆⡘⠀⠄⡀⢂⠄⡁⡐⢀⠆⡁⢌⠰⠀⠢⢈⠐⢄⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠆⣡⢈⡄⢣⠘⡜⢀⠀⡆⡐⣀⡧⠬⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠈⠉⠉⠉
⠀⠀⣀
⣿⣏⢭⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣶⣤⠠⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣌⠿⣿⡷⠐⣿⣿⠿⠷⢢⡀⠀⠀⢀⣤⡎⣿⣷⣮⣻⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⣰⡘⣿⣿⠿⣅⢰⡇⢠⣬⡿⠜⣿⣿⡿⢋⣽⡏⢾⣷⡾⣢⠗⢼⢱⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣶⡶⡈⠂⠁⠘⠚⠁⣶⢾⣿⠷⣙⢿⣿⣖⣴⡆⠁⠀⢸⢫⡷⢇⣀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⡿⢠⡉⡁⠀⠀⠀⠀⠘⠉⠁⠘⢿⣟⣫⣬⣿⡏⢸⣿⡿⢳⣝⣯⣩⣩⠥⢅⣂
⠀⠀⠀⠀⠀⠀⢸⡇⢍⣉⣼⣥⣬⣡⡀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⠛⣿⡅⠻⠗⣼⣿⣿⠉⠋⠴⠚⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠇⣼⣿⣿⣿⠇⣿⣾⡶⢤⣤⣄⣤⣶⣒⠿⣿⣟⣬⠏⠁⠀⠀⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣮⡉⢡⣺⣷⡿⢿⣷⣮⡍⣻⣿⣿⢂⡊⢉⠁⠀⠀⣠⣽⣭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡈⠋⠉⠉⠏⢿⡛⠁⠘⠻⠛⠻⠇⠀⠀⠀⠀⢨⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣴⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠿⠿⠿⠿⣿⣿⡿⠟⠋⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡽⠟⠛⠛⣯⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣦⣤⣄⣀⣀⣀⣀⣠⣤⣾⣿⣿⣿⣁⣀⣀⣠⣤⣴⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣵⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣤⣶⣶⣶⣦⣠⣾⣿⣿⠿⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⣾⡏⠀⠀⢀⣽⣿⣿⡋⠁⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄⣘⣧⣠⣶⣿⣿⠟⠹⣿⣶⣤⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣟⢿⡟⠁⠀⠀⠈⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣀⣤⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠟⠑⠀⠀⠂⠋⠀⡗⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⢿⣻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣦⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⡀⢤⣳⣿⠏⢐⣂⣀⢀⡀⣈⢮⣯⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⢹⣿⣿⢷⣯⣭⣷⣿⡿⠏⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠄
⠀⠀⠀⣠⡴⢉⠔⣿⣽⣯⡆⠈⠻⣿⣿⣷⣾⠿⠃⠀⠠⣌⣠⡼⠿⣿⠿⢿⣿⡀⣬⣯⣿⣿⣿⠛⠁⠀⠀⣠⣾⣿⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⡾⠋⡔⠁⢸⡷⣿⠁⠀⠀⠀⠀⠈⠀⠀⣰⣴⣿⡿⢿⣿⣧⠀⠑⣤⡶⣿⣷⣿⣆⠚⣿⠋⠀⠀⢀⠞⣿⣿⣿⡟⠛⠋⢀⣠⣤⣿⢇⣼⣿⣿⠏
⠀⠀⣿⣔⣆⢰⣟⣿⡟⠀⠀⠀⠀⣀⣠⣴⢿⣿⣿⣀⣀⣀⣈⡁⠃⠠⠘⣤⡿⠋⣿⣷⣿⠛⠃⠀⠀⠀⠀⠉⠉⠁⠀⣀⣀⣈⣁⣤⣷⣿⣿⠿⠋
⠀⠀⠫⢽⣛⣾⠿⠋⠀⣠⣤⠴⣿⣫⣽⡿⣿⣿⣽⣯⣭⣭⣭⠙⣶⣿⠟⠋⠀⠀⠙⠿⠿⡾⠶⠦⠀⠀⣀⠴⠾⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⡶⣟⣤⣴⣿⣿⣷⣾⠿⠿⠻⠿⠿⣿⡏⣸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⢊⣁⣤⠰⣶⣦⣶⣦⣄⡀
⠀⠀⠀⠀⠀⣰⠋⣹⣿⣿⣿⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣴⢦⡤⠀⠀⠀⠀⢀⡠⠔⠋⢀⣴⡆⣵⣼⣿⣸⣷⣿⣿⣿⣿⣿⣶⣦⣀⣀⣀⣠⣤⣤
⠀⠀⠀⢀⣾⣽⣾⣱⠾⠋⠀⠀⣀⣤⣠⣴⣤⠀⠀⠀⠀⠀⠈⠿⠉⣿⡆⠀⠀⠀⠊⠁⠀⢀⣴⡞⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠉⠛⠛⠛⠋⠁
⠀⠀⢀⠏⣼⣿⠟⠁⠀⠀⣠⡸⣿⣿⠞⣻⠇⣾⣿⠦⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠐⠛⠻⠒⠋⢻⣟⡷⢿⣯⡿⢿⡿⠋⠉⠀⠈⠁
⠀⠀⣼⣼⣷⠏⠀⠀⠀⢰⣿⣷⣶⢆⠁⠛⢼⡾⠁⣿⡿⢦⣤⢤⡖⠿⣧⣿⣗⡀⠀⠀⠀⠀⠀⠘⠉⠉⠛⠛⠛⡿⠇
⠀⠀⣿⣿⡿⠀⠀⠀⠀⡿⠙⢟⡁⠦⠀⠀⠀⠀⠿⠿⣃⣻⣿⣜⣛⠳⠒⣌⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠂⠄⢠⡀
⠀⠀⢻⣿⣷⠀⠀⠀⠀⠡⣿⣿⣿⢳⣆⡀⠀⠀⠀⠀⣹⣿⣧⣿⡇⣦⣤⣵⣿⣀⢀⡀⠀⠀⠀⠀⢸⢃⠀⠀⡀⠀⠀⠀⠀⠀⡀⠦⣅⡞⡁⠀⢀
⠀⠀⠈⢧⣻⣆⠀⠀⠀⠘⠷⠈⣰⣿⣿⣝⡶⢤⣶⡸⣿⣿⠈⠿⠘⠋⣵⣿⢻⡶⠟⢍⡄⠀⠐⠂⠿⠛⢄⣁⠑⢘⠓⢀⠂⣼⣹⣶⠿⠆⠙⠁
⠀⠀⠀⠀⠻⢿⣦⣄⠀⠀⠀⠀⠁⠀⠫⡿⠋⠸⣿⣃⡘⠋⠁⠀⣀⢠⣶⠍⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⠋⠁⠈⠀⢠⠕⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⡀⠀⠉⠀⠀⠀⠉⠀⠀⠀⢠⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⡞⠀⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⡀
⠀⠀⠀⠨⡻⢐⠀⠄⠀⡀⢀⡀
⠀⠀⠀⠀⠈⠂⠔⢀⡀⡄⠺⠷⣆⡀⢤⠦⠠⢠⠀⠀⠀⠀⠀⠒⠀⠀⠀⠒⠂⠀⠀⠀⢄⠤⠠⠤⠠⢄⣄⣀⡀
⠀⠀⢀⠀⢄⡀⠀⣤⣀⣦⢀⠀⠄⣿⠘⠂⢀⡀⠄⠀⠀⠀⠐⢃⠀⠀⢀⡀⢠⡄⠂⠰⠀⠀⢠⠀⢘⡀⢀⡉⠀⠁⠁⢲⠤⡀
⠀⠀⠀⠈⠉⠐⠚⠙⠒⠉⠙⠛⠂⠉⠀⠀⠀⠈⠡⠄⠀⠤⡀⠀⠐⢀⡀⢼⡏⢠⣼⣦⡄⢀⣉⣠⠼⠧⠨⠆⠠⢃⣠⠟⠐⢠⡦
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠾⣅⣀⠀⠠⡖⢳⣦⡗⣤⠀⠦⡄⠀⢁⡂⣳⡘⣿⣿⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡛⠛⣋⣥⣽⠒⠀⠐⠙⠠⠞⠳⣿⣷⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣠⣴⠶⠎⢘⣉⣿⣟⣛⠻⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⡿⠿⢿⣿⣟⠁⠂⠾⢟⡋⠉⠁⠁⡈⠁⣈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⣯⣴⠟⠉⠻⠒⠂⠀⠀⠀⠙⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣶⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠶⡿⠿⠿⠿⣷⣬⣾⣿⣿⡟⢻⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣵⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣷⡆⢀⡴⣿⠗⡈⢿⣶⣦⣾⣶⣼⣿⡟⢿⣿⣷⣄⠀⠀⠀⠀⠀⠙⣿⣿⣩⣮⣩⡆
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣀⣿⣴⠻⠞⠋⠈⠉⠙⠛⣿⣿⣯⠻⣿⣿⣿⣿⣷⣤⣀⠀⠀⣀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⢧⣿⣯⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⢨⣝⢛⣿⣿⠙⣻⣿⣿⣿⣿⣿⣿⣯⣾⡏
⠀⠀⠀⠀⠀⣿⣿⣾⣿⣼⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣛⣷⣾⣿⡿⢿⣿⡟⠛⠳⠟⠉
⠀⠀⠀⠀⠀⣿⣏⠻⣿⣥⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠎⠙⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣯⣜⢿⣿⣧
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣮⡛⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣎⢻⣿⣷⣌⣙⢶⣶⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣶⣶⣾⣿⣷⣯⣷⡎⢉⣿⢛⣷⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠿⣿⣿⣯⣬⣤⣽⣟⠾⢿⣿⣛⣿⣿⣿⣿⣿⣷⢿⡟⣿⣿⣿⣿⣽⣿⣿⣷⣤⣍⢸⣿⣿⡟⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡑⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣼⣿⢻⣿⣿⣿⣿⣿⣿⡿⣹⣿⠈⢿⣿⣿⣷⣮⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⢿⣫⣽⣿⣿⣭⣭⡭⠅⠠⠿⠟⢿⣷⣎⠲⣿⣿⣿⣿⣿⠿⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⠛⠿⠿⠿⠋⠐⠒⠂⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣾⣿⣿⡿⣿⣿⡆⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⣷⢻⣷⣿⣟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢿⣿⡆⣿⣿⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⢿⠏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣻⡿⢃⣤⣴⣶⣦⣽⡿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣿⣵⣶⠀⣿⣿⡟⣼
⠀⠀⠀⠀⠀⠀⣢⣿⡎⠙⢁⣤⡿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣝⣫⡿⠿⢿⣬⠿⠿⢟⣶⠶⢶⡄⠰⠶⠾⣯⣴⣿⡿⠇⠀⠈⠻⣿⣿
⠀⠀⠀⢀⠀⣿⣋⣿⣷⠶⢦⣤⡔⢻⣿⣶⣿⣷⣿⡿⣷⣄⣹⣛⡿⣿⣿⣿⠿⣿⣾⣿⣶⣶⣶⣿⣿⣿⢿⣿⣿⠟⣺⣿⣯⣽⣶⣶⡀⠀⠀⠉
⠀⠀⢠⣿⡇⢿⣿⣏⣤⡾⠿⠛⣡⣼⡷⠟⠛⠛⠛⠻⠛⠻⠿⠿⢿⣿⣿⣟⣤⣿⣿⠿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣾⣿⣷⣮⠛⠛⢠⣤⡶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⡿⢿⣷⣦⣶⣿⣿⣿⣿⠈⠁⢹⣿⡟⣛⣻⣿⣿⡿⠟⠁
⠀⠀⣩⣾⣿⣷⣤⡀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠆⠙⠫⣭⣤⡤⠼⣿⠿⠿⠛⠉⠁
⠀⢸⣿⣿⣿⣯⣽⣧⣽⡇
⠀⠀⣏⣿⣿⣿⣿⣿⡍⢷
⠀⠀⣹⡛⢿⣿⡟⢛⣰⠘⣷⣤⡀
⠀⠀⠘⢿⣦⣿⣿⣞⣋⣽⣿⣟⠻⡿⠷⣴⣤⣤⣤⣤⣠⣤⣤⣤⣠⣄⢠⣤⡀⢠⣤⣤⣄⣀⣀
⠀⠀⠀⠈⢿⡿⣿⣾⣿⣿⣭⣛⠻⣿⣿⠿⣿⣤⣤⣼⣿⡿⠻⣛⣇⣸⣷⣦⣄⢺⣿⣿⣿⡄⣄⣠⣿⣿⣶⡶⢠⣄
⠀⠀⠀⠀⠀⠙⢷⣹⣿⣿⣿⣿⡿⣫⡟⣿⣧⣴⣤⣭⣿⣧⣿⣿⣭⣿⣯⣭⣽⣿⠏⠛⣡⣼⢿⣿⠿⣿⣧⣤⣾⣿⣿⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠫⣿⣽⣾⣭⣤⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣤⣶⣿⣿⣯⣻⣷⣿⣿⣏⣰⣷⣿⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⣿⣿⣿⣿⣛⣿⣿⢏⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣧⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣏⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡶⢫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠆⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣼⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠿⢌⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡰⡇⢠⡟
⠀⠀⠀⠀⠀⢀⢀⣀⣤⡤⠴⠒⠲⠶⠶⠾⠶⠶⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⣬⠛⣦⠞⠁
⠀⠀⠀⣠⣤⡾⢿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⠟⠃⠀⣩⡾⠋
⠀⠔⠋⠈⠛⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠭⣭⣛⣛⣛⠛⠛⠿⠟⠛⢛⣛⣉⣡⣤⡤⠞⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠿⣿⣷⣦⣀
⠀⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄
⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣄⡀
⠀⠀⠀⠹⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠛⠳⠦⣄⡀
⠀⠀⠀⠀⠙⠿⣿⣟⡿⠷⠶⠒⣛⣛⣉⢩⣍⣉⡙⠻⢿⡶⣤⣀⣾⣷
⠀⠀⠀⠀⠀⠀⠈⠛⠩⠭⠭⠭⠥⠖⠒⠂⠀⠚⠿⣷⣦⡈⠪⡙⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣢⣶⣄⣀⡀⠀⢈⣹⣏⠢⡈⢢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⠛⠻⠿⢿⣷⡿⠿⠿⢷⡙⢄⠙⢮⡻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣖⣢⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠱⡀⠹⢦⡙⠦⣄⣀⠀⠀⠀⠰⣦⡀⠀⣀⣠⡴⠟⢋⣩⡴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠢⡀⠙⠳⣤⣍⣙⣛⣓⣒⣛⣿⡿⠟⣉⡤⠖⣋⣵⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠓⠤⣀⣈⣉⣉⣉⣉⣩⠤⠒⢋⣡⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⣭⣭⣭⣭⣥⣶⣶⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠠




⣿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣻⣟⣿⣿⠿⣿⣿⣿⣿

⠀⢀⣴⡶⠒⣶⣖⠢⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠛⠿⠛⠋⠁⠉⠉⠉⠓⠾⣷⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠲⡀⠀⠀⠀⣤⣶⠀⠀⠀⢀⡟⡽⠀⠀⠀⠀⠀⠀⢀⡀⠈⢿⣹⡄
⠀⠀⠀⠀⣀⣠⠄⢤⣤⣀⠀⠀⠀⠈⢳⡿⡄⠀⢸⣿⡿⠀⢀⡴⣫⠜⠀⠀⠀⠀⠀⠀⢰⣿⣿⠄⠘⣧⡇
⠀⠀⣠⣎⡴⠛⢋⡉⠛⢤⣷⡄⠀⠀⠈⣧⡧⠀⡾⣾⠁⠀⢸⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡙⣦⣼⢧⠇
⠀⢰⢉⡟⠀⣾⡝⠛⢳⡄⣧⣸⠀⠀⠀⣟⢹⠀⡷⣯⠀⠀⠈⠲⢍⣓⠢⢤⣀⠀⠀⠀⠀⢀⣼⣶⣭⡛⡄
⠀⢸⢻⡀⠘⣟⣆⡀⠘⣡⠛⡟⠀⠀⢠⠟⡏⠀⠹⣛⣦⡀⠀⠀⠀⠈⠉⠓⠦⡑⡆⠀⠀⣸⢴⠃⣠⠇⡄
⠀⠘⢾⣧⡀⠈⠻⠯⠽⠟⠋⠀⠀⣠⣿⡴⠀⠀⠀⠙⠳⡽⢆⠀⠀⠀⠀⠀⣠⢇⡍⠀⣠⠼⣾⢿⣵⠊
⠀⠀⠘⢦⣿⣤⣀⡀⡀⡀⣀⡤⢾⡧⠊⠀⠀⠀⠀⠀⠀⣿⢻⠀⢀⡠⢖⣫⠵⠋⢠⢊⡴⠋⠉⠛⢯⡷⢆
⠀⠀⠀⠀⠈⠓⠾⠻⠍⠽⠻⠒⠉⠀⠀⠀⠀⠀⠀⣠⣾⣹⠏⠀⠎⡴⠋⠀⠀⠀⠘⣌⠣⣄⡀⠀⠀⢹⠞
⠀⢀⣤⣶⡶⣶⣒⣶⣲⡦⢤⣀⡀⠀⠀⠀⠀⢠⢞⡷⠛⠁⠀⠀⢄⢳⣄⠀⠀⠀⠀⠀⠙⢲⣭⣝⣒⢯⡿
⠀⠘⠿⠿⠋⠉⠉⠉⠉⠉⠛⠶⣟⣲⣄⠀⠀⢞⣻⡁⠀⠀⠀⠀⠈⠓⠮⣝⠲⡀⠀⣴⣛⡿⠶⠛⠉⣧⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠳⡀⠈⠫⣽⡒⣦⣄⣀⠀⠀⠀⠈⢳⢹⢸⣤⣏⠀⠀⣠⠴⣣⠏
⠀⠀⠀⠀⢀⣠⢴⣶⣶⠦⣄⡀⠀⠀⠈⢿⣱⡀⠀⠀⠉⠛⠺⣯⣷⡄⠀⣠⠟⡜⠀⠓⢾⣛⣿⢵⡊⠁
⠀⠀⠀⡰⣯⡽⠋⣉⣉⡙⢧⡽⡄⠀⠀⠘⣧⣇⠀⠀⠀⠀⠀⣨⢧⡗⢰⣣⠞⠀⠀⠀⠀⡏⢿⠺⣏⣷⣄
⠀⠀⢰⢷⡏⢀⣎⡖⠉⢰⡄⣷⣿⠀⠀⠀⡷⡟⠀⢀⣠⠴⣞⡿⠞⠁⢡⣿⠀⠀⠀⠀⠀⠘⢮⡳⢄⣿⣾
⠀⠀⢸⢺⡇⠈⣏⢧⣀⣈⣴⢣⠏⠀⠀⣸⢳⠇⣴⣯⠿⠚⠉⠀⠀⠀⠈⠜⡄⠀⠀⠀⠀⠀⠀⣹⣦⢻⠃
⠀⠀⠈⢏⣷⣄⠈⠑⠚⠓⠚⠁⠀⢀⣴⣻⠎⠘⣷⡇⠀⠀⠀⠀⠀⠀⠀⠈⢾⠀⠀⠀⠀⠀⠞⠉⣹⠸
⠀⠀⠀⠀⠳⢽⡳⣤⣄⣀⣠⣤⡶⣻⠜⠁⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⠃
⠀⠀⠀⠀⠀⠀⠉⠑⠚⠓⠛⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠊⢑⡲⠦⠤⣒⠉⠉⠉⠉⠒⠢⣄
⠀⠀⠀⠀⠀⢠⠋⠀⠠⠒⠀⡀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⣎⠻⢖⣶⢤⣤⢀⡠⠀⠀⠀⠀⡠⠖⠒⠒⠦⡄⠀⠈⠆
⠀⠀⠀⠀⣸⠀⠉⠉⠀⠓⠷⠚⢀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⢆⠀⠸
⠀⠀⠀⡔⠁⡄⢀⠀⠀⠀⠀⠐⡹⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⢸⡀⠘⠡⠏⠀⡠⡲⠒⠚⠓⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡄
⠀⠀⠀⠝⡒⠤⠤⠜⢠⠁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⡴⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⢦
⠀⠀⠐⠌⠀⠉⠒⠊⠁⠀⠀⠀⢀⠔⠉⠉⠀⠉⢉⡱⠶⠋⠀⡠⠋⣀⣀⠤⣄⣀⡀⠀⡀⠀⠀⠀⢣
⠘⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡴⠚⠁⢀⡠⠔⠉⠉⠛⠢⣄⡠⠔⠚⠉⠉⠉⠉⠳⣄⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢠⠏⠀⠀⡔⠛⠒⠦⢄⡀⣠⠖⠉⠀⠀⢀⣀⣤⣀⡀⠀⠸⡄⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⠀⠀⠀⢇⠀⠀⣀⠴⠋⠀⠀⣀⣴⠊⠁⠀⠀⠀⠛⠀⢠⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠉⠉⠀⠀⢀⡤⠚⠁⠀⠉⠒⠤⠤⣤⡤⠴⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣵⣶⣶⣴⣶⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡿⢗⣶⢦⡄⠀⠀⠀⠀⠀⠉⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡅⢿⣿⣼⠏⣴⡖⣤⣀⢀⣀⠐⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣤⡐⣽⢳⣿⣿⣿⢛⣥⣤⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠻⠘⣯⢛⣟⠁⣿⣏⠻⠃⠉⠛⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠽⠃⡻⢿⣿⣿⣿⣿⣷⣦⣝⢦⡀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠘⣦⣳⡜⠦⣌⠏⢽⡷⣶⣬⣝⢷⣝⣆⣤⣄⠀⣴⣏⢩⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣻⣴⠀⠀⣨⣿⠁⠀⠈⠙⠲⠿⠿⠿⠟⢨⣿⠙⢷⡿⢸⣿⡟⠁⠠⣉⣦⡀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣊⠙⠟⠻⠿⠟⠁⠀⠀⠀⠀⠀⢀⣀⣬⣽⣯⣅⡰⠿⠀⠘⠋⠀⠀⠀⠈⡉⢀⡛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⣿⠆⠺⣤⡿⢇⣾⣿⣷⡄⠀⠀⠀⠘⡏⢦⠙⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡄⠀⠀⠀⠀⠀⠿⣴⢿⣿⡀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⣦⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⢉⢡⠐⢤⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠈⣧⠀⠁⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠀⠘⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠖⠒⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡜⠳⡄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⡞⠀⠀⠹⡄⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⠎⠀⠀⠀⠀⠘⣆⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢠⠏⠀⣸⠉⠉⡝⠀⠘⢆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣠⠃⠀⠀⡇⠀⣸⠁⠀⠀⠈⢧⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡰⠁⠀⠀⢸⠀⢠⠇⠀⣀⣤⠀⠀⢣⡀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⡼⠁⠀⠀⠀⡏⠀⣾⠴⠊⠁⡇⠀⠀⠀⠳⡀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⡜⠁⠀⠀⠀⢸⠁⣀⡤⣶⠀⢸⠁⠀⠀⠀⠀⠱⡄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⠞⠀⠀⠀⠀⠀⠿⠊⠁⠀⡏⠀⡏⠀⠀⠀⠀⠀⠀⠙⣄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡇⢸⢁⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢀⡞⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠹⡄
⠀⠀⠀⠀⠀⢠⠎⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠘⣆
⠀⠀⠀⠀⢠⠋⠀⠰⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠷⠀⠈⢆
⠀⠀⠀⢰⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣧
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢸⡀⢀⡇⢸⢠⠞⠉⠁⢸⡂⢀⡇⠀⢰⡀⢰⢃⡞⠉⢳⡄⡇⠀⠉⡟⠁⢰⢷⠀⣰⠊⠉⠀⣾⠉⠁⡇
⠀⠀⠀⢸⠉⠉⡇⢸⠘⣆⣈⣻⢸⡍⠉⡇⠀⠀⢧⡏⠘⣇⣀⣸⠃⣇⣀⠀⡇⢠⡟⠚⣇⢳⣀⣉⡇⣿⣉⡁⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠐⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂
⠀⠀⠀⠘⠮⠙⠛⠛⠛⢶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠀⠀⢉⠵⠂
⠀⠀⠀⢁⣀⠀⠀⠀⠀⠠⣄⣙⡟⢦⣀⡤⠤⠤⠴⠶⠶⠶⢦⣄⢀⣴⠒⠒⠒⠢⠤⠤⢤⣀⡄⣡⣎⣠⠄⠀⠀⠈⠉⣀⡀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠈⠁⢿⠉⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢶⠿⠿⠀⠀⠀⠐⠲⠇⠘⢶⡄⠀⢀⡀⠀⢀⡵⠂⠠⠖⠛⠂⠀⠘⢿⠿⡶⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠉⢿⠀⠀⠀⠋⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠙⣄⠀⢀⡿⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠀⢤⠂⢱⣼⣦⣤⣤⣴⣦⣄⣰⡀⠀⠀⢺⡿⠀⠀⢀⣆⣠⣴⣦⣤⣤⣴⣥⣮⣤⣿⢢⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⠾⣷⣠⠿⠟⣿⣷⣤⣀⣼⣇⣴⣤⣾⣿⠉⢻⡇⠾⠷⣖⠂⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠉⠋⢩⡿⠿⠿⢿⡟⠻⠉⠹⣿⠟⠛⢻⡿⠿⠛⠻⢯⡙⠉⠁⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠖⡋⠀⠈⡏⠀⠀⠀⢀⣰⣉⣀⠀⠒⠶⢦⡀⠀⠀⠁⠀⠀⣠⡤⠤⠄⠀⣀⣀⡀⠀⠀⠂⢹⠀⠀⠙⠒
⠀⠀⠀⠀⠀⢰⢁⠀⠀⠇⠀⠀⠀⠋⣤⠞⠛⢛⠋⠈⢻⣷⣄⣀⢀⣐⣊⣁⣶⡶⠒⡛⠛⠳⢬⢻⠀⠀⠸⠀⠀⡈⡆
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢸⠁⡀⢠⣼⠇⠀⠘⣿⠿⣿⣿⣿⠿⣿⡿⠀⠰⢧⡄⢀⠀⣧⠀⡗⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡸⡖⠁⢸⣧⡁⠂⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣼⡇⠈⣆⡿⢸
⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠛⠀⢻⠀⢉⡿⣹⠟⠒⠶⠶⣿⣿⣿⣶⣶⡖⣾⡟⣿⡟⠀⣿⠃⠀⠑⡜⣦⣳⢀
⠀⠀⠀⠀⠀⠀⠙⠱⡀⠀⠀⠀⠀⠀⠀⠣⠘⣧⣾⠀⣿⡗⢱⠃⠿⠻⣎⢹⡅⠸⣷⣿⢃⡜⡿⠀⠀⠁⠛⢠⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣀⠀⠀⠀⠳⣄⠹⣿⡀⣿⣷⡿⠷⣿⠶⢿⣾⣿⢀⣿⢿⣿⠄⠀⠀⣶⡀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡄⠀⠄⢀⠘⠳⡜⡓⢿⣏⠀⢀⠀⠀⠀⣹⡿⣾⢃⠞⡃⢠⠀⢠⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠀⠐⠀⠀⢹⡄⢃⣰⠀⢛⣠⢆⡌⢠⡏⢠⠀⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠦⠍⢁⠀⠒⠉⣚⠻⢾⣣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠈⢹⠇⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⢀⣤⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠆
⠀⠀⠀⠀⠀⠰⡄⢀⣀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⢀⠤⠶⠿⣧⡄⣤⠿⠗⠦⢄⠀⠀⠀⠀⠀⢲⣾⣯⣿⠻⣿⣦⡆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢨⣿⡄⣀⣀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣥⣤⣴⠆
⠀⠀⠀⠀⠀⠀⢣⡈⣶⠁⡰⠀⠀⠀⠀⠀⠀⠀⢠⣼⡟⢿⠻⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏⠉⢾⣿⣿⣿⡇⠀⠀⠀⠀⠈⠋⠉⠉⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣨⠟⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠈⠀⠀⠀⢸⣷⡾⠁⠀⠀⠀⠀⠈⠻⣿⠏⠀⠀⠀⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⣀⣼⣏
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⠀⠠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠸⠿⣄⣾⣿⣿⣿⣷⡤⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⣘⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣤⡆
⠀⠀⠀⢀⣀⡼⣇⣀⡿⣀⣸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠫⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣟
⠀⠀⠀⠘⠟⠀⠀⢿⣷⡿⠁⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡄⠀⠀⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣶⡦⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⣿⣿⡀⠀⠀⠀⠀⠀⠀⢰⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢣⡆⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠟⣹⠇⠀⠀⠀⠀⢠⡄⢸⣿⣾⣿⢀
⠀⠀⠀⠀⠀⢀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠾⣶⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣽⠋⠀⠀⠀⠀⠀⠀⠀⡀⣻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⡏⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢻⣿⣿⣿⠇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠉⠀⢛⣿⠟
⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠿⠿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⣤⣶⣤⡙⠄⠀⠀⠀⠀⠀⠀⠀⠇⠸⠀⡴⢃⠄⣀
⠀⠀⠀⢀⣄⠀⠘⢲⣿⣗⠃⠀⢀⡄⠀⠀⠀⠀⠀⢠⣾⠋⢀⣿⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⠛⣿⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⡈⠀⠴⢊⣡
⠀⠀⠀⠙⠛⠶⣤⣀⠶⢆⣠⠴⠛⠛⠀⠀⠀⠀⢠⡿⠃⠀⢸⣿⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣶⣿⠇⠀⠀⠀⠀⠀⠀⢠⣬⣄⠋⡀⠀⢈⡁
⠀⠀⠀⠀⠀⣠⣠⡽⠛⠯⣄⣠⠀⠀⠀⠀⠀⢰⣿⣁⣀⣀⣈⣉⣀⣀⣀⣻⣧⠀⠀⠀⠀⠀⢀⠀⠉⠳⣿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠐⠒⠁

⠀⠀⡄
⠀⠀⠀⠀⠂⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡀⠒⠠⢔⡀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣇⡄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⢁⠠⠀⠤⠤⠤⠤⡀⠀⠄⠀⠤⢈⠀⡄⠀⢀
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⣾⣧⠃⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢸⡥⠂
⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠼⣛⡽⠆⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⠀⢤⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⣀⡿⠛⠀⠈⠁⠀⠀⠀⣼⠶⠿⠿⣾⣧⡀⠀⢸⣦⠀⠷⠿⣏
⠀⠀⠀⠀⠀⠨⠉⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡀⠀⠙⢇⠼⣿⣿⡏⠁⠀⠀⠀⠉⠙⠂⠄⡀
⠀⠀⠀⠀⠀⠘⡁⠀⠀⡰⡏⠀⠀⠀⢀⣾⣋⠄⠀⠀⠀⢻⡉⠻⠙⠦⠀⠀⠀⠀⠀⠀⣀⠀⠑⢤⣀
⠀⠀⠀⠀⠀⣸⢀⠀⠀⠀⡷⠀⢸⠀⢸⢋⠴⢶⣦⣸⡄⠁⠀⢀⠀⠀⠉⠉⠢⣄⣠⣸⣯⣤⡈⣱⠋⡆
⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⠀⡈⡄⣿⡇⢳⣄⣾⠉⢻⡀⠰⡀⠑⡇⠀⠀⠀⠀⠀⠈⠉⠉⢹⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢳⠘⢾⣿⠊⡦⠩⢅⣒⣟⠷⣶⣤⣹⡄⠀⠀⠀⠀⠀⠀⣠⣼⠁⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠆⢀⡀⠀⠈⠈⠙⢻⠶⠛⠁⠙⣆⢰⠓⠂⠉⡛⠷⠤⠤⠔⠚⠉⣙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣡⣠⣀⣀⢹⡀⠠⠖⠂⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠣⣀⡴⠶⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡘⢦⡀⢄⣠⡶⠒⠛⠙⢷⣚⠛⠛⠒⣋⡿⠗⠒⡲⢤⣆⢠⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢷⣄⠉⠲⢿⡦⠀⠀⠉⠛⠶⠂⠀⠀⠶⠚⠉⠀⠀⠐⣿⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⡌⣻⣷⣄⠔⠀⠀⠀⠐⢦⡍⠀⠀⠀⠉⠀⢀⡀⠀⠀⠲⢄⠀⣿⣧⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣲⣿⣧⣤⣿⠁⢲⣦⠴⣿⢶⣄⠁⠀⠀⢰⠀⢸⢋⣴⡶⢦⡴⠀⣦⣙⣿⡯
⠀⠀⠀⠀⠀⠀⠀⢤⣬⢿⣿⠀⠋⢹⠀⡈⠃⠓⢿⠾⢻⡆⠀⠁⠀⠈⠱⠻⢬⠷⡺⠁⠀⠈⣤⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠶⢦⡞⣰⡷⠈⣲⠟⢤⡀⢠⠤⣄⣀⣠⣄⠠⠀⠙⢄⠰⣧⢫⡁⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⣿⠀⢹⢛⣿⢻⣶⣢⠈⠚⠲⡆⢠⢰⠄⠀⢴⣾⣿⠯⢻⡉⣿⡀⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢛⡿⠀⢘⡇⠘⢸⣿⣅⣁⡀⠀⠙⣻⠃⢀⣀⣠⣝⠛⠋⠯⢴⣿⠏⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠂⠹⣦⣾⣁⣀⣼⡿⣿⠉⣙⣛⡛⢛⣛⣋⠙⣿⡏⠙⢻⣶⣶⠏⢩⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠲⢾⠉⣹⠏⡀⣻⡝⡆⡿⠿⢿⡿⠟⣿⠸⣿⣀⢴⣉⢉⡤⠒⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣠⡙⠉⣷⢻⡷⣄⣀⣁⣴⣿⢰⡇⢁⣀⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣴⡟⣾⡇⣛⣛⣛⣃⣿⢷⠷⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⢘⠬⣛⠭⠀⠭⢝⣫⡎⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⢀⡤⠆⠀⠀⠲⠤⠹⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣁⡀⣀⣄⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡿⠛⠹⢧⡿⠅
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣴⣭⣿⣼⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠀⠀⣤⣶⣶⣦⡀⠀⣴⣶⣶⣦⠀⠀⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣅⢈⣿⣿⡄⠀⠿⣿⣿⡿⠁⠀⢿⣿⣿⡿⠀⢀⣿⣿⣏⢙⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡈⠁⠀⠸⠇⠀⠈⢁⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠙⣿⣷⣀⣠⣄⣠⣄⣀⣼⡟⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣤⣿⣿⡟⠁⠀⠀⠉⠛⠿⢿⣿⣿⠿⠟⠋⠀⠀⠀⠹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⢤⣶⣶⠤⠀⠀⠀⠀⠀⠠⢴⣶⣦⠤⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣏⠀⠀⠈⠀⢀⣠⣤⣶⣦⣤⣀⠀⠈⠁⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋

⠀⢰⣿⣷⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣄
⠀⠻⠿⠿⠁⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⣸⣿⣿⡦⠀⠸⣿⣿⡄⠀⢀⣠⣤⣶⣶⣶⣶⣤⣤⣀⠀⠘⠿⠿⠋
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠛⠉⠀⠀⠀⠀⠉⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣿⣿⣿⠛⠉⠛⣿⣿⣿⡿⠁⠀⠀⠈⢻⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⣷
⠀⠀⠀⢻⣿⡇⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⣿⠁⠀⠙⣿⣿⣿⡏⠀⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠻⣿⣦⣤⣶⣿⢻⡿⣿⣷⣤⣤⣴⣿⡿⠁⠀⠀⠀⠀⠀⠘⣧⣀⣰⡿⣿⢿⣷⣄⣀⣀⣴⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣾⣷⣿⣿⣿⡟⠛⢁⣤⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⠿⠛⡀
⠀⣠⣴⣶⣶⠄⠀⠻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠘⢿⣿⣶⠀⠀⣠⣤⣤⡄⠹⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠘⢿⣶⣦⡄
⠀⠙⠛⣿⠟⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠘⠛⠀⠀⠈⠛⢻⣿⠃⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠘⠋⠉
⠀⠀⠀⠀⢠⣤
⠀⠀⠠⣾⣿⣿⠀⠀⠀⣀⣀⣠⣄⣀⡀⠀⠀⠀⠀⢠⣶⣀⠀⠀⠀⠀⣀⣴⣶
⠀⠀⠀⠀⢈⣩⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠸⠿⠛⠀⠀⠀⠀⠛⠿⢃⣠⣤⣴⣶⣶⣦⣤⣤⣄⡀⠀⣼⣿⣤
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠛⠋
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢻⣿⣿⣿⣿⣿⠋⠀⠈⢻⣿⣿⣿⡿⠿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⡟⠉⠉⠙⣿⣿⣿⡿⠋⠉⠙⢿⣿⡇
⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠸⣿⡇⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣄⠀⢀⣸⣿⣿⣿⣀⣀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣶⣶⣿⣿⢙⣛⣿⣶⣶⣾⣿⠏
⠀⠀⠀⠀⠀⣀⣀⡉⠻⣿⣿⣿⣿⣋⣿⣍⣿⣿⣿⢁⣤⣤⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠋⠁⢰⣶⣦
⠀⠀⠀⠀⠺⠿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠘⣿⠟⠋⠀⠀⠀⢿⣿⣧⠀⠈⠛⠿⠿⠿⠿⠿⠛⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠄⠀⠠⠄⠀⠠⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣛⣖⢭⣍⢓⠀⠐⣲⣼⡖⢂⡀⠀⢀⡀⢉⣁⡀⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡵⣽⣿⣿⣛⡷⣦⣄⠒⢭⣽⣯⡿⣟⣻⡖⢂⢀⣾⣿⣿⣦⠲⠉⠁⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⠞⡾⣻⢿⣿⣯⣿⣵⣿⣿⢶⣿⣿⣿⣿⣿⡿⣦⣾⣿⣦⣿⣿⠀⠁⠀⠀⡀⠳⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠛⣏⡙⣩⣯⣟⣿⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⢿⣽⣽⣿⣿⣿⣿⣿⣠⣀⣤⡾⢿⢶⡏⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⣼⣷⣿⣿⣿⣾⣿⣿⣻⣿⣿⣿⣿⣿⣟⣿⣉⢠⣟⠻⣿⣷⣿⣫⡭⣽⣿⣿⣶⡶⣽⡷⢸⠊⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⡄⠈⠿⠅⣾⣿⣿⣿⣿⣿⣿⣿⡿⣽⣿⡿⠋⠉⠁⠀⠈⠉⠉⠛⠳⢽⣿⠻⣿⣿⣷⣿⣿⣿⡿⠇⠰⠸⠎⡳
⠀⠀⠀⠀⣀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡛⣿⠿⢻⣿⣿⣦⣷⣶⣶⣃⡟
⠀⠀⢰⣿⠿⣿⣟⣿⣿⣛⣿⡿⠿⠟⣿⣿⣟⣵⡛⠿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣁⣿⣿⣿⣿⣿⣷⣝⣿⢀⠇
⠀⠀⠀⢿⡻⢿⣍⢉⠛⡛⣟⡻⢍⡿⠟⢁⣈⣍⣁⣌⣉⡉⠷⠾⣟⢿⣿⣿⣿⣿⣿⡏⢽⡟⣟⢫⡲⠒⠶⢤⣄⡀⠀⠈⢫⣿⣿⣿⣿⣿⣿⡿⠇⠀⠸
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠈⠁⠀⠀⠀⠈⠛⠭⠙⠋⢽⣿⣷⣦⡀⢴⣮⣝⣿⣿⣿⣿⣆⢩⢄⣦⣶⢿⣻⣿⠅⠻⢷⣤⡈⢗⡠⠠⡝⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢄⢺⣿⣿⡿⣿⣆⠉⠸⣿⣿⣿⣿⣿⣄⣿⣿⣿⣯⢳⣿⠳⠀⠚⢻⣿⡆⣤⣶⣿⣿⣏⢋⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠈⠟⠳⡿⠟⣿⡀⢷⢹⣯⣿⣿⣿⣿⣟⣷⣿⢿⣿⠏⣥⣦⣼⣿⣿⡇⠹⣿⣿⣿⡿⣠⣄⡶⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⢁⠐⣘⣇⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⢂⣿⢿⣿⣿⣿⡇⠄⠈⡉⣻⠛⡉⠉
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣻⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠘⠿⠠⣿⠙⣟⣹⣿⣷⣿⣯⠙⠮⠧⣿⣿⡽⣻⣿⡿⢰⣿⠿⣿⣿⣦⣄⡀⠀⠰
⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⢛⣫⣽⣿⣟⣿⣿⣟⣿⣷⢠⡀⠀⠀⠀⠀⠈⢰⣿⣿⣿⣿⣿⣿⣦⡁⠄⠽⢿⣾⣿⡿⢁⡜⢻⣴⣿⡇⠀⢉⣷
⠀⠀⢀⣼⣿⠧⠼⠿⣿⣏⣽⡿⠬⣝⠎⣻⣟⢿⣿⣿⣿⣻⣿⠛⣦⡀⠀⠀⠀⣘⠻⠿⠿⠟⣻⣿⣿⣿⣆⠐⠰⡽⠟⢡⣤⣤⣄⡾⠋⠻⠷⠋⠁⠀⠂
⠀⢀⡿⣿⣗⣚⢞⢞⡻⠛⠿⣙⡳⢌⣻⡛⠿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣦⡀⠀⢩⡄⣧⢠⣾⣫⣿⣿⣿⡟⣲⣾⣷⣦⣻⣯⣿⡟⢧⣄⠀⠀⠀⢠⠂
⢀⡾⠓⢉⣿⣇⠠⡤⡜⠉⡒⢾⠛⠀⠂⡉⣍⢏⣛⣻⣿⣽⣿⣿⣿⣿⣿⣿⣆⡀⠁⣧⡈⣻⣵⣿⡿⠋⢳⣿⣿⣿⣿⠗⢻⣯⠀⠠⢹⡃⠀⣀⠊
⢸⠗⠟⣛⢩⡜⠾⡶⠀⠀⠀⠀⠀⠀⠀⠉⠏⠀⠀⠚⠿⢭⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣅⣊⡅⠀⠀⢹⣯⣙⣧⣿⡇⠀⠙⠿⠛⢉⡐⠀⠀⠈
⠾⠀⠘⠄⠀⠂⠀⠀⠀⠀⠀⢀⠔⠑⠘⠲⣆⡰⠀⠀⠀⠋⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣝⡻⢲⣥⣮⣉⡛⠛⠛⠀⡀⣀⡄⠔⠋⠡⢷⡆
⡗⠺⠄⠀⠀⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠙⢶⡼⠿⡆⠀⢨⣩⡭⢷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣏⠙⠙⣗⡋⠉⢳⣍⢀⠀⠘⠛⠛⠁
⠳⠪⢤⠀⠛⠀⠀⠀⠀⠀⠉⠧⠀⠀⠀⠀⠀⠀⠀⠘⢦⣠⢀⢈⠋⠁⠈⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡾⣿⣗⣁⢻⣧⠂
⢠⠝⠀⡀⠀⡀⠀⠀⠀⠀⠀⠂⣷⣄⡀⠀⠀⠀⠀⠀⠀⠙⢷⣎⢢⣄⣀⠙⢂⣭⢿⢿⢯⣛⣿⣿⣿⣻⣿⡡⠈⡉⢻⣧⣼⡟⠂
⠀⢠⠂⠁⠄⠀⠀⠀⠀⠰⠒⠁⠤⠉⠟⠷⠶⣤⣤⣤⣠⣤⣤⣼⣿⣿⣿⣤⡀⠄⠂⠛⠛⠹⠭⠿⠿⠿⠋⠐⠊⠄⠁⠈⠀⠀⠀⠠⠀⠈⠀⢠
⠀⠀⠶⠃⠀⠖⠀⠀⢀⡆⣠⣤⣴⣶⣤⠖⢶⡦⢴⣾⠤⢤⣄⣈⡁⠈⠙⠻⣿⣶⣷⣤⣤⡠⣀⠀⠀⢀⣀⠀⠀⠀⠀⢤⡄⠀⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠐⠀⠀⠀⢸⢀⣧⣿⣾⣷⣿⣿⣤⣿⣧⣸⡇⢰⡆⢀⡈⠙⠛⠢⣄⣀⣈⢙⡛⡿⠿⡿⣷⣾⠿⠶⠶⠒⠚⠯⢤⣀⠠⠂⠐⠀⠀⠀⠀⠂
⠀⠀⠀⠲⠀⣠⢼⡞⢏⣹⣉⡧⠷⠿⠛⠛⠛⠿⠿⢿⣾⣧⣴⣯⣀⣄⠀⠀⠉⠙⠻⢏⡉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣿⠞
⠀⢤⠀⣄⠾⠉⢣⠰⠚⠙⠢⠄⠀⣀⣀⢀⠄⠠⣀⣠⣀⣉⡻⠿⢿⣧⣼⡃⣤⠀⠀⠀⠈⠛⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠁
⢀⢠⠒⢈⣵⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠉⠛⢿⣤⣟⠀⡄⠀⠀⠀⠈⠙⠷⠶⠦⠦⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠤⠦⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⣤⣀⠀⡀⠀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠀⣀⣄⡀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢧⢀⠆⠀⢁⣾⣿⣿⣿⣶⣤⣀⣴⠆⠀⢰⣤⣀⣤⣾⣿⣿⣿⣷⡘⢀⢰⠀⡾
⠀⠀⠀⠀⠀⠀⠘⣾⢞⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢀⢤⡘⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⡿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡦⠣⠀⢄⠹⠿⠿⡿⢟⠏⡡⢣⣶⡹⡌⢛⡻⢿⠿⠿⠋⡤⠀⠎⢼⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠴⠃⠀⢱⣿⣿⣿⡌⠀⠙⠦⠀⠀⠈⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠶⠶⠂⣀⡀⠤⠀⠀⠀⠭⣿⠍⠀⠀⠀⠄⢠⣀⠐⠶⠶⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⣶⣿⣿⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢿⣿⣷⣶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠹⡿⠲⡖⡶⠒⡞⠉⡟⠉⣷⠒⣖⢲⠒⢿⢋⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣷⢸⡦⣧⠤⣧⣀⣧⣀⣷⠤⣷⢼⠇⡟⠘⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢄⡹⠖⢧⣧⠀⡇⠀⣯⠀⣧⠀⣧⠼⠰⢃⠴⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⡄⠀⠀⠉⠈⠒⠉⠂⠁⠋⠀⠀⣰⡷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢤⣀⣀⣄⣀⣀⠄⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⠤⠀⠀⠀⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡤⢀⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⡄⠀⠀⠈⠉⠉⠓⠒⠂⠀⠉⠉⣁⣀⡄⣴⠋
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠢⠴⠤⣠⣃⡼⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⢧⣙⣄⡤⠦⠔⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠠⢄⠀⠙⡙⠀⠀⠈⠢⠈⢄⠠⠄⡠⠀⠖⠁⠀⠠⠋⠋⠀⣠⣴⠖⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⢀⣣⣴⣠⣄⣠⠀⠀⠀⠈⠀⠀⠁⠀⠀⢀⣄⠀⢠⡀⠘⠛⠁⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⡀⠛⢛⣦⠀⠀⠹⡏⠀⠀⣴⡛⠛⠀⠀⢿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠍⠁⢺⣿⠓⠦⠴⠳⡖⠙⣿⡷⠈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠙⢧⠈⠀⠀⠀⣽⠋⠷⠤⡴⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⡷⢶⣴⡐⠈⠀⠀⠀⢸⠀⠃⠀⠀⠠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠋⡟⢁⢈⡀⠀⠀⢘⡄⠩⣄⡐⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡶⡻⠁⡬⠥⢤⣀⣘⡡⠀⡾⢵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣱⢄⡀⠀⠀⠒⣉⢵⡱⠷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠉⢑⠚⡛⡋⠁⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠀⠤⠴⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠟⠟⠛⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠿⠿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠈⠙⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣾⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⣀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣦⡀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣄⣦⣦⠀⣠⣶⣿⣶⡤⠶⠶⠶⠶⠶⠞⠶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣷⣿⣿⠿⠋⢀⣾⣿⣿⣿⣿⣾⣿⣻⣷⣦⡄⣳⣶⠄⣀⡼⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠁⢠⡇⣼⣿⡿⣟⣛⣛⠛⠻⠽⠟⣁⣨⠭⢖⣫⣵⡾⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⠋⠀⠀⢸⣷⣿⡟⠉⠉⠉⣉⠭⢔⣚⣭⡵⠶⣻⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠟⠁⠀⠀⢀⢸⣿⣿⣷⣄⠀⠀⠀⠀⢀⣀⡤⠖⡫⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡏⠀⠀⠀⢀⣾⣸⣿⠃⠈⠉⠉⠛⠛⠛⠛⠉⣠⡞⣱
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠹⣇⠀⠀⢀⣾⣿⢻⣿⡄⠀⠀⠀⠀⠀⠀⣠⣾⣟⣴⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠹⡄⠀⣾⣿⣿⡜⣿⣷⣄⠀⠀⣀⣴⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⣀⣨⣭⣭⣥⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⣿⣿⣿⣛⠛⠛⠊⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠘⠟⠛⠛⢷⣄⠀⢹⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠙⠄⠘⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⡄⠀⠰⠂⠀⠀⠂⠀⢀⣀⠀⠀⠀⠄⠲⠀⠀⠀⠀⡀⠀⣠⠆⠀⠀⠆⠀⢀⠀⠀⢀⠀⠀⠀⠰⠀⠀⡠⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⡀⠁⠀⠀⠁⠓⠀⠀⠁⡄⢀⠆⡰⢡⣷⣊⡄⠀⠔⠀⢀⠀⠀⠀⠄⠈⠀⠐⢤⣄⠀⠀⠀⠐⠈⠁⠀⢀
⠀⠀⠈⣄⡠⠁⠀⠀⠉⠀⠃⠀⡄⠀⡇⢰⠃⠈⢠⣶⡪⠥⠂⠀⠃⠀⠠⠀⠐⠀⠀⠔⠠⠠⠄⠀⠀⠉⠓⠤⡄⠀⠀⠀⠉
⠀⠈⠁⠀⠁⠀⠠⠤⠀⡀⢰⠀⠙⠸⠀⡀⢠⣶⠋⠄⠄⠒⠂⠐⢁⠀⠀⠀⠠⠆⠀⠀⠀⠠⣠⠎⢀⠀⠀⠀⠀⠺⢆⠀⠈
⠈⠀⣄⠀⠀⠀⠀⢰⠀⡇⠈⠀⡆⢸⡰⠷⠇⠀⠀⠀⠒⠒⠀⡀⠀⠀⢀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠘⠀⡤⠀⢠⠂⠘⠃
⠀⠀⠀⢄⠀⡆⠀⢀⡀⠀⢰⠀⣧⣈⣉⠓⠂⠀⠈⠒⠒⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠠⠀⢀
⠀⡀⠰⠸⠀⢇⠀⠀⡇⢻⣘⠁⠀⠀⠀⠁⠀⠉⢉⡁⠀⣀⠀⠀⠄⠀⠀⠀⠀⠟⠂⠀⠀⠐⠀⠀⠀⠈⠂⠀⠀⠀⠂⠐⣄
⠀⠓⠀⠀⠀⢸⠀⠀⠣⠤⣬⡉⢀⠁⠒⠦⠄⠀⠞⠙⢦⡈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⣇⠘⣤⣘⣓⠒⠂⠠⠤⣈⠛⢂⡤⢨⢥⠛⡀⣁⠐⠆⠀⠀⠀⢀⠆⢀⠀⠀⠊⠀⠀⠐⠒⠀⠀⠿⠄⠀⠀⠀⢁
⠀⠀⠀⠀⠐⠓⠒⠀⠠⢄⣈⡑⠂⠀⠀⣉⠝⣇⠈⠆⠁⣨⠀⠘⠃⠈⡡⠄⢀⠀⠀⠈⠀⣀⠄⠀⠀⠀⠛⠀⠑⡄⠀⠀⠈
⠀⡠⠂⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠈⠁⠀⢠⠀⠺⠀⠀⠀⠀⠀⠼⠃⠈⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠐
⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠘⠀⠘⠀⠀⠀⡠⠚⠀⠀⠀⠀⢀⠠⠋⢀
⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⣢⡄⠉⠀⠀⠀⠀⠄⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⢆⠀⠀⠀⠀⣰⠦⠤⠄⠀⠀⠀⠤⠀⡀⠄⡀⠀⠀⠊⠀⠀⠉⠀⠀⠀⠈⠁⣀⠜⢏
⠀⠀⠀⢀⣀⠄⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⣠⡜⠓⠀⠂⠀⢀⣀⠐⠀⠐⠃⢀⡠⠀⢰⠀⠁⠀⢀⡠⠀⠀⠰⠀⣠⠒⠀⠘
⠠⠤⠒⠊⠉⠀⠀⠈⢀⠀⠈⠀⠐⠄⠀⣴⣋⡁⠀⠀⠀⠀⢠⠅⠀⠀⠀⠀⠀⠀⠀⠈⠃⣠⠔⠋⠀⠂⠀⠔⠀⠀⠤⠀⠈
⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠂⠀⠀⠰⠯⠀⠀⠀⡀⠀⡋⠀⠀⠂⠀⠀⠱⠀⠀⢠⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⡲⠄⠀⠰
⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢣⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠦⡀⠀⠊⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡀⠜⠊⠀⠀⡀⠀⠀⢀⠔⣴⣾⣿⣿⡿⣿⣿⣿⣿⣷⣦
⠀⠀⠁⡀⠀⣐⠆⠂⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠡⠐⠀⠊⠀⠀⢆⠀⠀⠈⡠⠀⠁⢸⣿⣿⡿⢿⡷⣿⣿⡿⣿⢿⣿
⠀⠀⠃⠀⠀⠀⠿⠰⠙⠀⠒⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠱⠀⠀⠀⠀⠀⢸⣿⢸⡇⣻⡇⢹⠃⡆⣿⡸⣿
⠀⠀⠀⢀⠠⠊⠀⠄⠀⠀⠀⠀⠀⢠⠀⣤⢀⣀⣐⠀⠀⠀⠀⠀⠀⠈⠁⡠⢐⠊⠔⠃⠀⠘⣿⣼⣧⣻⣴⣾⣴⣇⣿⣣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠠⠀⠌⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠡⡀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠈⠀⠀⠀⠀⠠⠊⠉⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡕⠂⠀⠀⠀⠀⠀⠀⠐⡀⠀⢀⢸⠀⠀⠀⡠⢤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡣⡴⠃⠀⢀⠜⢁⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⣠⠔⠃⠀⠀⠀⢀⠇⠣⡠⢖⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⠤⡄⢸⡁⠀⠀⠀⠀⠀⠀⣸⠄⠀⠀⠧⣀⡠⠤⡄
⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⠀⡇⠘⠲⠂⠀⠣⡀⠀⠀⣠⠴⠊⢁⠞⠒⠶⠶⠒⠒⠴⠃
⢀⣀⠀⠀⠉⠒⣌⠢⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠓⠉⠀⠀⡠⠋
⢏⣈⣉⣒⣒⡤⠞⠀⠘⢦⠤⠴⠚⠀⠀⠰⡉⡱⠀⠀⠀⠀⠀⠉⠒⠢⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠑⡆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢈⠇⢀⡰⠒⠒⠤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡞⠁⢀⡠⢚⣀
⠀⠀⠀⣖⠉⡷⠒⠉⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⣎⢙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⢒⡯⠒⠉
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠈⠁⠀⠀⠠⠖⡄⠀⠀⠀⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⢀⡠⠔⠊⢀⠀⠈⠢⢤⠔⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠘⠤⠜⠀⠀⠀⢎⢉⠆⠀⠀⢀⢤⡀⠀⠙⠋⠀⠀⣰⠃⠉⠉⠉⠉⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠃⠀⠀⢀⡀⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡨⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠒⠒⠉⠉⠉⠉⠁⠀⠀⠈⢹⠀⠀⢀⢤⡀⠀⢀⣀⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⢸⡀⢠⠏⠀⠈⢣⡡⠤⠇⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠈⡆⠀⠀⠀⠀⢀⠔⠋⠀⠘⢆⠀⠀⠀⠱⡀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣀⠔⠁⠀⠀⣠⠴⡣⠖⢲⢠⠶⢌⢣⡀⠀⠀⢇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠳⠤⠃⠀⣸⢹⠀⠀⣇⣨⠆⠀⡘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢎⠀⠀⠀⠀⠀⢠⠃⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠞⠀⠀⠀⠀⣠⠋⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠉⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠉⠉⠐⠛⠓⠉⠁⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠤⣀⣀⣀⣀⣀⠤⠴⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠾⢿⡠⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢷⡄⡰⢷⣇⠴⠐⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠈⠑⠦⣄⡀⠀⠀⣠⠶⠒⣚⠛⠛⠒⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⣄⣀⣉⡉⠉⠹⠄⠀⠀⠀⠀⠀⠀⠀⢈⣉⡷⠤⢈⠙⢦⣤⣤⡶⠚⠃⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⢠⠇⠀⢰⠀⠀⠀⢀⣀⠤⣖⠋⢹⡄⠘⣠⣿⠀⠀⣖⠙⠀⢰⣿⣿⡇⠐⢋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣼⣀⡤⠚⠉⣆⠀⠉⠀⣸⣇⡀⠈⠻⣦⠌⢋⣑⡀⠀⠉⣉⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⣤⠟⠁⠋⠀⣸⣿⣆⠀⢀⡉⠉⢟⡨⠀⠀⠀⡼⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠁⠘⠳⣶⠂⠀⠈⠻⢿⣄⠈⣡⡔⠉⠀⣈⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠫⢷⡄⠀⠀⠾⠁⠀⠚⢂⣀⡤⠖⠋⠻⣀⠀⠫⣀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⡒⠛⣿⡄⠘⠷⢀⣜⠁⠐⠒⢶⠒⠒⠛⠉⠁⠀⠀⠀⠀⠈⠓⠦⣌⡓⠦⠼⣇⣴⠖
⠀⠀⠀⠀⠀⠀⣠⠶⠋⢿⠄⣈⣣⠤⠞⠒⠊⠉⠁⠈⢳⡀⠠⡈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⣄⠼⢦⣄⡀
⠀⠀⠀⠀⢠⠞⠑⢇⡤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠘⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡇⢸⡆⠀⠓⠃
⠀⠀⠀⣰⠛⣢⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢰⠧⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡠⠊
⠀⠀⣟⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣇⢠⠊⡴⠋
⠀⠀⡏⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡁⢺⡄⠀⡀
⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠹⡍⢯⡳⠟
⠀⠀⢷⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣧⠈⠓
⠀⠀⠘⣆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠘⣆⠣⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠓⠮⠭⡿⠿⠛⠁

⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠉⠉⠉⠉⠙⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢡⣶⣷⡀⠀⠀⠀⠀⣰⣿⣦⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡟⣰⣿⣿⣿⣷⡄⠀⠀⣰⣿⣿⣿⣷⡘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣔⠿⠛⠛⠻⢗⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⣿⣿⣿⡏⣴⣿⣷⡌⢿⣿⣿⣿⡇⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⠶⣾⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠹⠿⠿⠃⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠉⠉⢨⡝⠻⣿⠀⠀⢸⡿⠛⣽⠀⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⣾⣶⣾⣦⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣷⣄⢹⣿⣿⣿⢁⣼⡟⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⢠⣾⣿⣿⣿⣿⣧⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣺⣿⣿⣿⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣭⣙⣛⣛⣩⣥⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠟⠛⢿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣤⠀⠀⠀⠀⠀⠀⢠⡜⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣷⣿⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣼⢁⡾⠀⣤⣦⡄⠸⣆⢻⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢻⡘⣧⠈⠻⠿⠏⢀⡟⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠷⠈⠀⢠⣆⠀⠈⠰⠟⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⠁⠿⠋⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣸⣿⡀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢿⠀⠀⠀⠟⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠃⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁



⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⢻⡋⡹⠉⠉⠛⠟⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠝⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠑⠚⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠈⣧⠀⠀⠒⠂⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⡠⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣧
⠀⠀⠀⠀⠀⢻⣤⠤⢄⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⠹⣆⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡄
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⡀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠹⣆⠀⢀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣧⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⣀⠀⢠⠤⢄⠀⠈⠙⢶⣄⠀⠀⠀⢻⡄⠈⠂⠀⢳⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠉⠛⠺⢦⣔⠀⠀⠐⠤⠘⢷⡄⠀⠘⣧⡠⢄⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠹⣆⡠⠙⢦⡀
⠀⠀⠀⠀⣠⡴⠛⠉⠉⡉⠙⠳⢦⡀⠀⠈⠳⣄⠀⠀⠀⠈⣿⡄⠀⢹⡆⠀⠃⡀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢻⣄
⠀⠀⠀⣰⠋⠢⠀⠀⠈⠁⠀⠀⠀⠙⠷⣤⣠⡿⠀⠀⢀⠀⣹⡇⠀⠀⢿⡀⠨⡈⡂⠹⡄⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠘⣧
⠀⠀⢠⡏⠁⠈⠀⠀⡤⠶⢤⣀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢢⣿⠇⠀⠀⠘⣧⠀⠈⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣷⠔⠦⡘⣇
⠀⠀⠘⡇⠀⠀⠀⢸⡇⠀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠁⣿
⠀⠀⠀⢻⠈⠠⠂⠈⠧⢤⡀⠀⠀⠈⠛⠷⣦⣴⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⢄⠀⠙⣦⡀⠀⠀⠀⢀⣠⠏⠀⠀⢠⡇
⠀⠀⠀⢸⠀⠠⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⡹⠛⠒⠚⠋⠁⠀⢊⣨⠟
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠒⠂⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠹⣄⣶⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⠶⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂

⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⠶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢆
⠀⠀⠀⠀⠀⠀⢹⡀⡄⠀⠀⠀⠀⠀⢀⣤⣠⣿⣿⣿⣿⣷⣠⣄⠀⠀⠀⠀⣼⠛⠏⠀⠀⠀⠘⠟⢷⡄⠀⠀⠀⠀⠀⠀⡼⠁⠈⢦
⠀⠀⠀⠀⢀⡆⣼⣇⣃⠀⠀⠀⠀⠀⠈⠛⢛⡏⢙⣿⡉⣻⠛⠋⠀⠀⠀⢰⣿⣶⣆⠿⠂⠺⢆⣶⣿⡇⠀⠀⠀⠀⢀⡞⠀⢀⡴⠉⢣
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣀⣤⣿⡿⠾⢿⣯⣄⣀⠀⠀⠀⠀⢿⡉⣩⡐⠶⢆⣌⡉⣻⠇⠀⠀⠀⢀⠞⠀⢠⠎⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠛⠋⠈⠻⠶⠛⠈⠙⠋⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠿⠋⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⣴⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⢠⢃⣀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⡰⠃⢈⣏⠀⠳⡀⠀⠀⠀⠀⠀⠀⢀⢎⡴⠛⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⣘⣾⣿⣿⣿⣦
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡰⠁⠻⠏⠈⠿⠃⠱⡀⠀⠀⠀⠀⢠⠎⠘⢿⣅⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⢹⣿⠋⠈⣿⡿
⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠛⠀⠀⠀⠀⠛⠒⠀⠀⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠋⠁

⠀⠀⠀⢀⣴⣶⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣾⣿⠿⣿⡿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⡆⢀⡴⣩⠗⢄⠀⠀⠀⠀⠀⡴⠁⢀⣠⣄⡀⠈⢆
⠀⠀⠀⢸⣿⣤⡞⢳⣤⣿⠁⠀⠀⠀⠀⢀⣾⣦⠈⠟⢁⣾⡆⠀⠀⠀⠀⢠⠇⠀⢨⢁⠋⢀⡥⠞⣉⣇⠀⠀⠀⡜⢡⣦⣿⣿⣿⣿⣴⠌⢧
⠀⠀⠀⠀⢠⣹⡿⢿⢃⡆⠀⠀⠀⠀⠀⠸⣿⡿⠃⡀⠙⣿⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠠⡨⠒⠋⠁⡿⠀⠀⠘⡄⠀⣀⣳⣾⣷⣞⣀⠀⣠⠃
⠀⠀⢀⣦⣈⡛⠿⠾⢟⣁⣴⡀⠀⠀⠀⠀⠙⠁⣼⣿⣆⠈⠁⠀⠀⠀⠀⠀⢣⡰⣿⡿⠀⠉⠨⠉⡹⠃⠀⠀⠀⠘⣌⠛⠉⠳⠞⠉⠛⡰⠃
⠀⠀⢀⣀⣈⡽⠗⠺⠯⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠥⠤⠤⠔⠊⠀⠀⠀⠀⠀⠀⠈⠦⠤⠤⠤⠤⠔⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀
⠀⠀⢠⠊⠁⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⢀⣧⣴⣶⣶⣶⣶⣶⣆⠀⠘⠀⠀⠀⠀⠀⢀⣶⢸⣿⢀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠀⠄⠀⡴⠉
⠀⠸⠈⠉⠉⠉⠉⠉⢻⣿⣦⠀⠃⠀⠀⠀⣴⣿⣷⣾⣯⡾⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⠙⣿⣿⣿⡦⠀⠀⠀⠀⠚⠓⠀⠈⠁⠀⠀⠠⠄
⠀⠀⢣⠀⠀⠀⠀⠀⠈⠛⠛⡿⠀⠀⠀⠀⣿⣿⠿⠻⣿⣿⣆⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠐⠐⠈⡄⠐
⠀⠀⠀⠑⠠⠀⢀⡀⠀⠤⠊⠀⠀⠀⠀⠀⠈⠻⣦⣤⠿⠿⠋⠀⠀⠀⠀⠀⠀⠹⣿⣶⣤⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠆⠀⠈

⡿⠿⠿⠿⠿⢿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣛⣿⣿⣿⢿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠏⠀⠈⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠈⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠁⠀⢠⣾⣿⣿⣦⠀⠀⠹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠁⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⡀
⠀⠀⠀⣰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡀
⠀⠀⣰⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀
⠀⣼⣿⣿⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣿⣿⣿⡄
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⣰⣿
⠀⠀⠀⢹⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠴⠶⠶⡖⣞⣿⣶⣶⣦⣄
⠀⠀⠀⠀⠻⣿⣿⣟⢿⢟⡻⢛⣛⣛⣿⣽⣥⣷⣶⣿⠿⠿⠿⠿⠿⠿⢿⣿⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠛⣛⣛⣋⣭⣭⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠛⢉⣉⣁⣍⣉⡉⡙⡛⠛⠻⠿⠿⠛⠛⢉⣼⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣴⣾⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⢷⠿⢧⣼⠿⠿⠛⠁
⠀⠀⠀⠀⠀⢠⣿⢣⣶⡿⠋
⠀⠀⠀⠀⠀⣿⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⢿⣿⡜⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣟⣋⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⡽⣷⡀
⠀⠀⠀⠀⠀⠘⢿⣷⡌⠻⢷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠛⣋⣴⡼⣛⡿⠟⠛⠉⠉⠉⠁⠈⠉⠻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⠻⣿⡾⡄⠉⠛⠿⠿⣶⣶⣶⣶⠿⠿⠟⢋⣩⣤⣜⣿⡭⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠧⣯⣄⢂⠀⠀⠀⠀⡀⣀⣀⣠⡴⠿⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⢉⣦⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣒⣒⣬⣭⣴⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡝⡿⢿⣿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢿⡯⣵⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢲⡄⣼⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣾⠋⠉⠁⠀⠀⠀⠀⠀⢠⣀⣠⣄⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣧⣤⣤⡴⣴⢦⣏⣿⡄⠀⠠⣽⣿⣀⣠⣿⣻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢷⣼⡷⣿⢽⣷⣭⢥⣾⣿⡿⠖⠛⠋⠉⠉⠉⠉⠛⠻⢧⣤⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡻⣿⣭⣟⣶⣿⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⡿⣏⠻⢍⣼⣿⣿⣯⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣷⢢⣤⠀⠀⠀⠀⢠⣟⣛⣧⡼⢋⣶⡞⢻⣿⡷⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⣿⣧⠀⢀⡴⠷⣹⠗⡘⢸⠿⢦⣼⣾⠻⣿⣿⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⣶⠟⣳⣟⣠⠾⣃⣍⣜⣿⠟⠁⠀⠈⠉⠛⠻⣿⣿⣶⡶⠆⠀⠀⠀⠀⠀⢀⣄⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⡯⡑⢫⡴⢛⣤⣖⣥⣿⠟⠁⠀⠀⠀⠀⠀⠠⠾⣿⣿⣿⠻⠆⠀⠀⠀⠠⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠛⢟⢿⡻⣿⢝⣤⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡑⠁⠂⢂⠈⠇⣺⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⠀⠠⣲⡨⣈⣽⡿⠁⠀⠀⢙⣿⣾
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣤⣵⠾⠿⠛⠉⠀⠀⣀⡠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⢿⣿⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠾⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡾⠀⠀⠀⠀⣀⠤⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣟⠀⣠⣾⠧⣴⣿⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠁⢰⣿⡀⣤⣾⡫⡾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣧⣀⠀⡇⣿⣇⣩⡙⠗⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⣯⠛⣿⣍⢘⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣁⢺⣇⢀⡿⠁⣭⠏⠀⠀⣶⣧⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⣷⣶⠿⢿⣿⣧⠈⣿⣿⣦⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡛⠉⢷⣶⣶⠙⣿⡿⠋⠙⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣶⣾⡯⢠⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣤⣼⣯⣤⣬⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢸⣿⡇⠉⣻⠃⠀⠀⠀⠀⢀⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣈⣿⣇⡈⢿⠀⢀⣠⣤⣶⣿⣿⣷⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠈⢋⠉⣿⡏⢁⣼⣶⣿⣿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣸⣿⣿⣶⣿⣿⣿⠻⣧⣀⡿⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡿⠿⠛⠛⠳⢦⡉⣿⣇⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⣟⠛⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠙⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣶⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⢀⣾⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠳⣤⣀⣀⣀⣤⡞⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠿⠀⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⠭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣾⠕⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣴⡋⢉⢿⣿⣦⢠⠀⢀⡀⣰⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣎⣽⣿⣷⣶⣦⠿⣿⡀⠻⠟⠋⠙⣿⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⣿⣿⣫⣾⣿⡍⣝⡷⡄⠀⠀⠀⣜⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣸⣯⣼⣿⣿⠟⠍⢿⣯⣷⣷⠢⣄⣤⣧⣄⣤⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⡿⢿⣿⣿⡓⣤⣼⣟⣿⣅⣤⣿⣩⣽⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠳⣄⠲⣿⢿⣿⣦⢬⣿⣍⡻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠠⢞⣿⡿⠀⠀⠀⠉⣍⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡊⠽⣷⣦⣤⣾⣿⣿⣷⣬⣶⣆⣺⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢉⡟⠛⢹⡿⣏⠀⠈⠙⣿⣿⣿⣿⣿⣯⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣤⣾⣤⠶⠛⠁⠨⡟⢦⣄⣈⣻⣿⣿⣿⡇⢼⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡴⠦⣄⠱⣌⣠⣿⠿⢿⡟⢿⣿⡙⢿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠒⠰⠿⠷⠿⠟⢤⣤⣶⣼⣿⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢶⡻⠮⣅⣈⣀⣠⣤⣴⣶⡾⠛⠿⠅⠉⣛⣿⣿⡟⠀⠀⠀⠀⠸⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠈⠉⠁⠀⠀⠀⠀⠉⠙⢻⣷⣦⣌⣿⣿⣿⡇⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣬⣽⣶⣶⣀⣐⣨⣤⣶⣿⣿⡿⠋⣿⣿⡟⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢹⣿⣿⣿⠟⠀⢰⣿⡿⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣀⣀⣿⡿⠁⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠋⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣠⣤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⠂⠀⠐⠐⠀⢸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⢿⣿⣿⣿⣿⣿⣾⣿⡗⠄⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣷⣦⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢀⢀⡜⣶⡿⠀⣒⢀⡀⠀⠀⠡⣹⡇⠀⠀⠀⠀⠀⣠⣤⣼⣿⣿⠹⣿⣿⢷⣿⣭⣵⣶⣶⠿⠁⠀⢀⢜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⢀⡤⠜⡡⣺⣻⣯⣧⠀⠹⣿⣿⣽⣶⣿⠟⠀⠀⠠⡐⣀⡼⣿⣿⣿⣿⣿⡀⣍⣯⣾⣿⣿⠿⠋⠀⠀⢀⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠀⠁
⠀⠀⣴⡟⡰⠋⢀⣷⣼⠏⠁⠀⠀⠀⠈⠉⢉⢀⣠⣴⣶⣾⢿⣿⣇⠀⠡⣐⡮⣿⣷⣿⣆⠼⣿⠟⠀⠀⠀⡰⣻⣿⣿⣿⠿⠿⠃⢀⣀⣼⣿⢇⣾⣿⣿⠇
⠀⠀⡷⣀⠇⡀⣞⣿⣟⠂⠀⠀⠀⠀⠀⣀⣴⡿⠿⠯⠁⠀⠈⠙⠉⠇⢢⠸⣥⣿⠟⣿⣷⣶⠿⠖⠀⠀⠀⠁⠒⠋⠉⠀⠀⠀⠀⣋⣉⣹⣵⣿⣿⡿⠋
⠀⠀⠻⣻⡿⣾⣿⠽⠁⠀⣠⣀⣴⢞⣛⣿⣿⣿⣿⣿⣭⣭⣩⡍⢱⣴⣷⡶⠛⠁⠀⠹⢿⣽⣦⣥⣦⡀⠀⠀⣀⣤⠶⠾⠿⠿⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠈⠉⠉⠀⢀⣴⣾⣋⣀⣾⣿⣿⣽⣾⣿⠿⣿⣿⣿⣿⡇⢸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠛⣁⡠⣤⣤⣠⣤⣀⡀
⠀⠀⠀⠀⠀⢀⡴⠛⣽⣾⣿⣯⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⢿⣇⣧⢠⣀⡀⠀⠀⠀⠀⠀⢀⡠⠖⠃⢠⡦⢙⣬⢶⡏⣿⣽⢿⣿⣿⣶⣦⣤⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣴⣟⣴⢚⣭⠿⠋⠀⠀⢀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠛⡟⠛⣾⡄⠀⠀⠀⠔⠊⠁⠀⢀⡶⣿⣷⣞⣿⣿⣿⣯⣿⣿⣷⣿⡍⠉⠉⠁⠓⠾⠿⠿⠟⠋
⠀⠀⠀⡼⣹⣿⣿⠟⠁⠀⠀⡀⣾⣿⣍⠿⣿⢩⣶⣦⡄⠀⠀⠀⠠⠖⠉⠁⠀⠀⠀⠠⠶⢶⡬⠿⠿⣿⣿⣽⣿⣿⣿⣿⡿⠛⠋⠉⠻
⠀⠀⢰⢡⣾⡿⠁⠀⠀⠀⣾⣿⣬⣍⠙⠰⢇⣬⡿⢩⣧⣤⡀⢀⢀⡤⣶⣆⣶⡤⠀⠀⠀⠀⠀⠀⠀⠟⠛⠺⠿⠿⢶⣷
⠀⠀⣿⣿⣿⠁⠀⠀⠀⢰⠻⣿⡿⢡⠃⠀⠀⠋⢰⣾⣿⡯⣾⣿⠾⢷⣏⠝⡸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⡤⠀⡀
⠀⠀⢿⣿⣿⠀⠀⠀⠀⠸⣰⣶⣾⡶⣥⠀⠀⠀⠀⠉⠀⢷⣾⡋⣷⡇⣁⣐⢟⣿⠃⠀⠀⠀⠀⠀⠀⠠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡴⠀⠀⠀⠈⠃
⠀⠀⠘⡏⢿⡄⠀⠀⠀⠀⣮⡛⢛⣱⣻⣗⢄⣀⢀⡀⣐⣾⣿⠻⣟⣰⡟⣛⣿⣿⣑⣲⠂⡀⠀⠀⠀⣰⡁⠁⠀⠨⠀⠀⡀⠠⢀⡞⣦⣩⣿⡐⡶⠂
⠀⠀⠀⠘⢾⣿⣄⠀⠀⠀⠈⠀⠚⠛⢻⣿⣷⠌⣿⣷⠎⠻⠷⠀⠀⠀⢀⣛⡟⠈⠓⠖⠑⠂⠀⠀⠀⠀⠀⠀⠚⢦⣠⡴⠤⠂⠞⠓⢛⡋⠉⠀⠄
⠀⠀⠀⠀⠀⠙⠻⢷⣦⣀⡀⠀⠀⠀⠀⠹⠄⠀⠙⠛⠿⠀⠀⠀⢠⣶⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠲⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠘⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⡀
⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⠚
⠀⢸⣿⣿⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠
⠀⠘⣿⣿⣋⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠘⣿⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠸⣿⠀⠀⠀⢸⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣶⣿⣀⣀⣠⣼⣿⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⠟⠉⣿⠋⠉⠉⠉⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢀⣴⣾⣿⣄⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⠶⣿⣶⠟⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠁⠀⠀⠀⠀⠐⢄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⣿⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠁⠀⠈⠀⠀⠀⠀⠀⠀⠑⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠿⠃⠀⠀⠀⣸⣿⣿⣿⣿⡿⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⣿⠿⠣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⣾⣾⢶⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣾⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣤⣤⣤⣴⣶⣶⣿⡿⠿⠛⠛⠉⠉⠉⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⣄⢌⠀⠰⡘⢃⠀⠀⠋⣰⠟⣱⡿⣻⣿⡟⠀⠀⣰⣿⣿⣿⣄⠀⠘⢀⢀⡄⠐⣶⣌⣿⣿⣿⠿⠟⠋⠁⣿
⡮⣑⣇⠀⠀⠈⠀⠀⠘⠃⠀⠋⣈⣼⣿⣀⣀⣀⣿⣿⣿⡻⡿⢷⣦⠀⠳⣿⣶⣾⣝⣿⣿⡇⠀⠀⠀⠀⠻
⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠈⠋⠿⣿⣿⣿⣿⡿⣿⡝⢿⡄⠈⡯⠳⠀⠀⠉⠛⠿⢿⣿⡏
⠀⠉⠢⠄⣈⢦⡀⠀⠀⠀⠠⠒⠁⠈⣦⣍⠙⠻⣷⠈⢿⡄⠀⠀⠀⠀⠤⠀⠀⣠⣄⠀⠈⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠒⢦⣄⡀⠀⠘⢯⣝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⢙⣧⣤⣄⣠⣼⣿⡿⢷
⠀⠀⠐⠂⠀⠀⠀⠀⠀⢠⣐⡽⠃⠀⠀⠀⠙⠻⠇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠲⢿⣿⣿⡷⢞⡫⠅
⠂⠀⠀⠀⢐⣤⠀⢀⢨⣦⣿⡆⠀⣠⣶⣬⡄⣀⣀⠀⣀⠠⠔⠒⡫⠉⠁⠀⠀⠀⠀⠈⣉⠭⣭⡶⠋⢴⣿
⣴⠋⠀⡄⢹⣿⣦⣇⢿⣿⠟⠀⠀⠒⠂⢀⡬⢔⣈⠝⠀⢀⠀⠎⠀⠀⠀⡀⠀⠀⠸⣿⣿⡿⠋⠀⠀⣰⠟
⠇⣀⣴⣃⢀⡉⠙⠛⠈⠁⠀⠀⠀⠀⠐⠻⠶⠿⠟⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠦⡄⠀⠀⣼⡇⣿
⣾⣿⣿⡿⡼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣿⢠⢀⢠⣀⡄⠀⢧⡀⠀⣿⠀⠀⠘⠛⠛
⠟⠸⠁⠃⠀⣿⣿⠀⠞⠙⠙⢣⣶⣶⣤⣀⠀⠀⠀⠀⠀⢿⢏⡿⣿⢸⡘⣿⠷⣦⢲⣽⣷⣾⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠍⠁⠀⠤⠤⠞⠫⠪⠫⣾⠇⠀⠀⠀⠀⡀⣤⣾⣷⢿⠜⠗⠀⠀⠘⢷⣗⣽⠇⠀⠀⠀⠈⠻
⠀⢀⠀⠀⠀⢀⡀⠀⠀⠤⣤⣾⠛⠉⣟⢿⡗⣄⡀⣠⡹⡌⣇⣿⣾⣦⠀⠀⠀⠀⠀⠀⣈⣠⠤⠀⠐⠂
⠀⢸⡇⡀⠀⠀⠀⠀⠀⠐⣛⣛⣡⣴⣷⣿⣛⠌⠻⢿⣷⠸⠸⢹⡿⠏⠀⠀⠀⢀⣤⣾⣿⠋⠙
⠀⠀⡇⠈⢢⠀⠐⠶⣬⣴⣒⡶⣶⣖⢰⡀⠉⠳⡄⠈⠁⠀⠀⠈⠃⠀⠀⠀⠸⡿⢏⡝⠁⣀⡍⣀⣠
⠀⠀⢡⠀⠈⠂⠀⠈⢺⣿⣿⠝⠛⠉⠈⠳⠦⠀⠀⠀⠀⠠⠀⠀⠀⡔⠁⠀⣘⢤⠩⠂⡴⢶⣞⣉⡀
⠄⠀⠞⡀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣦⠀⠀⠀⠀⠀⠘⣀⠴⠊⠀⠀⠠⢀⣀⣠⡿⠋⣠⢀⣼
⠄⣶⠀⢷⢹⡿⠀⣀⣴⣿⣦⡀⠀⠀⢠⡿⠭⢉⠀⠀⠀⠀⠀⠴⠋⠁⠀⠀⠀⠀⠀⠛⠻⢿⣵⣾⠷⠟⠁
⢸⡟⡄⠀⠻⢁⣾⡟⢸⣿⣿⣧⣀⣠⣤⣿⣿⣿⡦⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠛⠿⣦⡀
⠎⣿⠡⠄⠠⡿⠘⠁⠼⠿⢻⡿⢿⢿⢿⣿⠛⠻⠿⢿⡕⣠⠞⠀⠀⠀⠀⠀⠀⠻⣦⡈⠿⡿⣷⣶⣶
⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠀⠀⠀⠀⠉⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣶⣶⣶⣄⠀⠀⣠⣶⣶⣶⡄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠙⠛⠛⠉⢀⡀⠉⠛⠛⠛⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⣴⣿⣿⡦⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⢢⡀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⢀⡔⠁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠈⢳⡀⡇⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⢸⢀⡞⠁⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠸⠥⣙⡢⠔⠓⠤⠚⠓⠤⠚⠲⠔⣋⠬⠇⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣠⠦⢄⣀⠉⠙⠲⠤⢄⣠⠤⠒⠋⠉⣀⡤⠴⡄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡹⠖⠾⣅⠀⠀⣀⡭⠖⠲⠤⣀⡈⠉⠒⠶⢭⣀⠀⠀⣨⠷⠲⢏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠊⠉⣀⡀⠤⠒⠋⠉⠈⠙⠒⠤⢄⣈⠉⠙⠁⠀⠀⠀⡇⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠈⢳⠀⠀⠀⢰⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡆⠀⠀⠀⡞⠁⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠘⢤⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⡤⠃⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢧⡀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣀⣀⠀⠀⢀⡼⠁⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢈⣿⠞⠋⢉⠀⠀⢀⣠⣤⣨⣅⣤⣄⡀⠀⠀⡉⠙⠳⣿⡁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⣀⣴⠟⢁⣴⢿⣷⠖⠛⠉⢞⣡⢭⡭⣌⡳⠉⠛⠲⣾⡿⣦⡈⠻⣦⡀⠀⣠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⢋⡴⢻⡇⠀⠀⢀⣘⠛⠛⣩⠤⠤⢍⠛⠛⣃⡀⠀⠀⢸⡟⢦⡙⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠞⠉⣰⣿⣇⡀⠀⠊⢁⡠⠖⠒⠒⠒⠒⠲⢄⡈⠓⠀⢀⣸⣿⣆⠉⠳⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢀⡄⠉⣽⡏⠻⡄⠀⠺⠀⠀⣠⠔⠢⣄⠀⠀⠗⠀⢠⠟⢹⣯⠉⢠⡀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⣾⢇⣾⠁⠀⠗⢢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠲⠀⠈⣷⡼⣷⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢸⡏⠈⠙⠲⠄⠀⠙⢺⣿⣏⣀⠀⠀⠀⠀⣀⣹⣿⡗⠋⠀⠠⠖⠋⠁⢹⡇⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢸⣇⣠⡤⡠⢞⣣⣤⠄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠤⢤⣜⡳⢄⢤⣄⣸⡇⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣠⣴⡟⣇⠀⢰⣿⣦⡻⢄⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⡠⢟⣴⣿⡆⠀⣘⢻⣦⣄⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣬⣧⠻⠆⢸⣿⣏⠻⢦⡀⠀⢠⡄⠀⠀⢠⡄⠀⢀⡴⠟⣹⣿⡗⠰⢟⣼⣥⣿⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣯⣍⡛⠛⣻⣿⣓⠦⠀⠀⠈⠻⣤⣤⠟⠁⠀⠀⠴⣚⣿⣟⠛⢛⣩⣽⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠹⣿⣿⠿⠳⠶⡶⢤⣴⣦⡴⢶⠶⠞⠻⣿⣿⠏⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣀⣿⣿⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⣿⣿⣀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣆⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⣰⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡏⠖⡀⠀⠀⠀⠀⢀⠲⢹⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣟⣉⣉⣉⣙⣩⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⡤⠠⣤⠠⣤⠀⠀⣠⠔⠐⢤⠀⠀⣤⠀⠠⡄⠠⣤⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⣼⠃⠀⠀⠀⠀⠀⣿⠀⢠⠀⠀⣿⠀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⢻⡄⠀⠀⢻⠃⠀⣿⠀⠈⠀⠀⣿⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠐⠛⠀⠀⠙⠒⠄⠚⠀⠀⠛⠒⠒⠃⠐⠛⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠲⡄⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣱⡀⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⡠⠶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢤⣷⡆
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⣈⡷⣈⡿⠸⠏⠀⠀⠀⠀⠀⢦⠀⣸⠃
⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠆⢃⠉⣥⣙⡹⢿⡾⡋⠀⠀⠀⠀⠀⠀⠈⢧⢸⣠⠃⣀⡤⠦⠀⠀⠀⢀⡀⢠
⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣋⡥⣨⠶⣿⣿⠺⣍⠠⠙⢬⣦⠀⠀⠀⠀⠀⠀⣨⡿⠿⠯⠵⠦⠀⢀⣤⠾⢻⣁⣾
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣏⡙⠯⡖⢳⣬⠐⢨⣙⢿⣷⣈⠄⢰⠓⡀⠀⠀⠀⠀⡝⣯⣋⠀⠀⠀⢀⢺⡏⢰⣾⠀⣿
⢠⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠐⣩⡼⠉⠈⡿⣿⣯⣝⢮⢷⣿⣶⢦⡳⣝⣆⠀⠀⠀⠑⢟⡿⡗⢀⡀⡀⢎⠻⣿⣥⠶⡇
⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠂⡟⠀⠀⠀⠀⢸⢸⠗⠆⠒⡻⣇⣲⣝⢨⠛⠥⣲⡤⠤⣬⡅⠹⠿⢉⡭⢉⢄⠒⠊⠙⠀⡇⠀⢀⡤
⠀⢰⢁⣦⡀⠀⠀⠀⠀⠀⠀⡟⡎⣷⠀⠀⠀⠀⠀⢰⡟⣼⠈⢯⡳⡕⢈⡻⣿⢳⣬⣐⠀⣤⣤⣤⠠⠐⠘⢉⡴⣧⡎⠀⠈⢳⢰⢃⡴⣃⣤
⠀⠀⠀⢧⡻⠓⡆⣠⡤⣤⡀⠷⠖⠃⠀⠀⠀⠀⠀⢠⣤⢏⡀⠀⠁⡿⣀⣙⡮⣝⠻⠟⢽⠷⠶⠶⣚⡉⢄⡧⡘⠛⢧⣦⡀⣰⣿⣿⡯⢥⡄
⠀⠀⠀⠀⠙⠲⠇⣻⣤⣷⡭⠗⠣⣄⡀⠀⠀⠀⠀⢬⣄⣾⠁⠀⠀⠈⠋⢬⣝⣲⠮⢥⣥⣬⣼⠯⠥⠄⠘⠁⠀⠘⠋⣵⢚⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⣭⣿⣾⡶⠛⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠖⢊⡽⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠐⢈⠉⠩⡉⠐⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠤⡈⠢⡀⠱⠀⠃⠀⡇⢠⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⡀⠤⢂⠌⠀⠀⠀⠀⠀⠀⠔⠁⠀⢄⡈⠓⢄⠘⠆⠘⠀⢀⡀⠀⢀⠀⠐⠠⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⠄⠀⠐⠈⠐⢄⠀⠀⠀⠠⢊⠀⢠⡢⡀⠈⢦⠈⠂⠀⠠⠂⠁⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢐⠢⡀⠁⢀⢱⣀⠔⠡⠀⠀⠂⠀⡌⢆⠀⠁⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢀⣑⠒⠗⠂⠟⠉⠀⠁⠀⠀⠐⠂⠀⠆⠀⢀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⣴⠆⠈⠀⠀⠀⠀⠀⢠⡄⠀⠤⠀⠐⠋⢀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢀⠔⠁⣠⠄⠁⠀⠀⠀⢠⡤⠀⠋⠀⢀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐
⠀⠀⠀⠀⠀⠌⠀⡀⠁⠀⠀⠀⠀⡤⠈⠀⠀⠀⣾⠁⠀⠈⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅
⠀⠀⠀⠀⠰⠀⣸⠃⠀⠀⠀⢠⡄⠀⠀⢀⡞⠀⠁⠀⡀⠂⠀⠄⠃
⠀⠀⠀⠀⠂⠀⠙⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⢠⠎⢐⠤⠀⠀⡁
⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠋⠀⠀⡠⠊⠀⡁⠸⠂⠄⠐⡀⠀⠁
⠀⠀⠀⠀⡀⠀⣿⣶⡄⠀⠀⠀⠾⠀⠀⡐⠁⠀⠈⠒⢁⠂⠂⢁⢀⠄
⠀⠀⠀⠀⢁⠀⠀⣤⠀⠀⠀⠀⢠⡄⠀⠁
⠀⠀⠀⠀⠀⠆⠸⠿⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠈⡄⠀⠯⠀⠄⠀⠁⣀⠀⠦⠤⠀⠀⠀⢤
⠀⠀⠀⠀⠀⢀⠌⢢⠀⠂⠀⠴⠆⠈⢀⠐⡃⠀⠀⠀⠁⡆
⠀⠀⢀⠄⠂⠁⠰⡀⢡⢸⡂⠀⠁⠀⣨⡅⠀⠀⣰⢀⡀⠧⢂⡆
⠀⠀⠂⠘⢠⠂⠁⠀⢀⠘⠃⣀⠙⠓⢀⡀⠺⠷⠈⠂⠀⠐⠄⡀
⠀⠀⡀⠀⠠⠠⢒⠄⡆⢀⣄⠁⠀⣤⠞⠷⣤⠀⢰⣿⠈⠈⠂⠘
⠎⠀⢀⠀⠀⡘⡐⠂⢁⠘⠻⠆⠴⠰⡄⠀⢥⢠⠀⡌⠈
⠀⢀⠂⡎⠱⣀⠄⠀⠀⠢⠀⡔⠀⣃⢰⡀⡆⠂⣠⠇
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠑⢄⠈⢧⠀⠇⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠆⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡛⣟⣚⠻⠿⣶⢦⣄⣶⣤⣤⣀⠠⣤⣄⣠⣤⣶⣦⡽⠿⠹⠿⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡾⣿⣿⣿⣿⡷⣬⣫⡝⣽⣯⣵⣹⡓⠈⣉⡼⢯⡙⢷⢤⡀⠀⠀⠀⠈⠉⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠾⠿⣹⢳⣯⣿⣿⣿⣿⣿⣿⣿⣝⣹⠿⣟⣭⣍⣿⣧⣵⡈⢩⡯⡿⢮⣯⠽⢖⣒⡒⠠⣄
⠀⠀⢀⣠⣤⣔⡲⠞⡿⣿⣻⡧⠾⠿⠿⠾⠵⠿⠛⠛⠉⡁⠙⠻⣦⣬⡴⢏⣽⣿⣦⣙⣾⣦⣙⣂⣹⢶⡯⢽⣽⡆⢧⣿⡄
⠀⣔⣏⣭⡿⠟⣛⣾⣗⢻⣧⣀⠀⠀⠀⠀⠀⠠⣸⣿⣯⣽⡏⠓⣢⣌⠉⠁⠉⠉⣉⣥⣤⢬⣿⣯⣍⣀⣿⣦⣈⣿⣲⡴⣺
⠘⡘⠪⠾⠧⠛⠛⠚⠻⣷⢠⣌⢳⡄⠀⠀⡀⠀⠀⣹⢭⣽⣿⡿⢾⣫⣷⣤⣶⢟⡽⣟⣒⣶⣩⡿⣷⣿⣌⡻⢿⣾⣟⣘⡟
⠀⠈⠁⠒⠀⠁⠀⠀⠈⠒⣾⣙⣧⣻⢆⠀⠈⠚⢩⣴⣩⢟⠱⢦⡥⠉⢋⣿⣿⡿⠚⠉⠨⣿⣼⡁⡏⡋⠉⠙⠓⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⣷⠉⣜⡷⠦⠄⣈⣉⣉⣤⣶⣿⠀⠀⢸⣠⣏⡀⣀⡀⢴⣋⡺⠏⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣿⡉⣷⢻⣸⡿⠿⢿⣿⣿⠀⠀⠈⣌⣹⣧⠜⣃⣒⢫⣗⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠋⠛⠿⠓⠚⠙⠉⠲⠄⣀⡈⠁⠻⠟⣋⣼⡯⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⡀⢀⡀⠠⢤⠤⠤⠤⠴⣤⣤⣀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠀⢉⡁⠀⠈⠉⠉⠭⡭⢉⣓⣒⠯⢽⣺⠦⣦⣀⣀⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠏⠁⠀⢀⠉⠓⠀⠀⠀⠀⠐⠒⠒⠒⠒⠲⠷⠀⢻⠭⣗⣺⣽⣿⣆⢸
⠀⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢶
⠀⠀⠀⠀⠀⡀⠛⣍⠈⠁⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠄⠀⠀⠀⠀⠀⠀⢢⡀⠘⣿⠁⠀⠔⠁
⠀⠀⠀⠀⡀⠀⠀⠸⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠻⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⣿⠁⠁⠀⠊
⠀⠀⠀⠀⠙⠀⠂⠠⢸⡂⠠⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⣿⡤
⠀⠀⠀⠀⠀⠂⣀⠀⠌⣇⢀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡄
⠀⠀⠀⠀⠀⠀⠁⢀⠀⢿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠚⠲⠒⠓⠚⠀⠛⠒⠓⠒⠒⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠐⠠⡀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⡄⠀⠀⣸⠅⠀⠀⠀⢤⠠⠀⠙⠀⠀⠁⠉⠃⠀⡀
⠀⠀⠂⠀⠀⠀⢲⡆⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⡼⠃⠀⠀⠉⠀⠀⠀⠳⢤⡄⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠠⠤⠤⢴⣾⡿⠃⠈⠉⠂⠀⠁⠀⠀⠀⢠⡼⠄⠀⠀⠀⠀⠀⠈⡓⠀⠈⠈⣙⡦⣄⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣟⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⡀⠐⠉⠹⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢁⣧⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠻⣽⢟⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡘⡯⢚⣚⢄⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⠸⢰⢫⢮⡃⡾⣿⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⣷⣔⠘⣼⣿⢫⣿⣞⡟⠀⠀⣀⠀⠈⡀⢀⠙
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⢌⣿⣱⣌⣛⣛⣻⠟⠁⠀⠀⠁⢲⣴⢷⣮⣀⠄⠸⠂
⠀⠀⠀⡦⡀⠀⢋⠃⠀⣠⠀⠀⠀⠙⠶⢖⣿⢻⢇⣀⠀⠀⠀⠀⠀⣉⣻⣿⢤⡀⣄
⠀⠀⠀⠀⠙⢦⣨⣤⡴⠈⠀⠀⣠⣾⡯⡕⣽⡷⣡⢶⡭⠖⠤⣤⠞⡽⠂⠀⠀⠀⠁
⠀⠘⠃⠐⠒⢸⣽⠛⡇⡀⢀⣴⢋⠸⣧⢟⣽⣵⣳⡯⡭⠽⠂⣦⠞
⠀⠀⠺⠃⠈⠁⠀⠈⠙⣩⣭⣧⠞⢗⣦⡝⣧⣻⣼⡖⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠸⣏⣺⣽⣿⢻⡳⢯⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡣⢾⣾⣸⣷⡬⠻⢼⢷⡄⠀⠀⠀⠀⠀⠀⠀⢫⢳⠀⣰⠎⢠⡋⠋⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣯⠘⣛⣿⢿⣾⢟⣜⣿⡼⢦⠀⠀⠀⠀⠀⠀⠀⠸⡶⡻⣴⠋⠀⠀⡀⠀⠀⠀⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡱⠭⢯⡝⣣⠣⣬⣽⢗⣣⣯⣷⡀⠀⠀⣠⢴⡶⣦⣆⣩⣿⠒⠒⠀⠛⠁⠀⠀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢿⠞⢱⣿⢿⣿⣷⣹⣍⢿⣜⣧⠀⢰⢳⣫⡞⠁⠁⠀⠉⠣⣢⡄⠀⠀⠀⡼⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢞⢹⢮⣴⣯⣹⣮⡻⣟⣷⣩⡽⠆⠾⡿⠿⣧⠀⠀⠀⠀⠀⠈⠀⢀⣤⣾⡾⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢬⡓⣾⣏⡯⣽⢿⣿⠷⡷⠴⠶⠲⠖⠋⠀⠀⠀⠠⠴⠶⠻⢻⣿⣿⣷⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠿⢯⣦⡊⣿⣛⣿⣛⣿⣟⡿⢭⣭⡿⢿⣭⡛⢶⡿⠿⢋⣿⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣨⢝⣮⠭⣭⡉⠻⢿⡔⢿⣽⣩⣝⢾⡝⡆⣯⣛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣻⠿⠶⠿⠚⠛⠀⠀⠀⠉⠒⠚⠿⠷⠚⠃⠐⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⣢⣤⣀⣈⣻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⣼⡱⢆⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠊⢁⣬⠛⣟⠃⠿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡥⠁⢐⠹⠀⠐⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠋⠉⠉⠛⠛⢿⣿⣿⡿⠛⠋⠉⠉⠙⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀⠀⠀⠀⠀⣠⣿⣿⣀⠀⠀⠀⠀⢀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠛⠉⢻⣿⣿⣿⠇⠀⠀⠸⣿⣿⣿⡟⠉⠛⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⣸⣿⣿⣿⣦⣤⣤⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣦⣼⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⣿⣿⣿⣿⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠛⠉⠀⠈⠙⠻⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⢤⠠⠠⠄⢴⣒⣖⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡘⣌⡃⠀⠀⣯⡷⣿⡀⣀⣢⣼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣿⣽⢮⣦⣯⣴⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢊⢉⣉⣼⠋⠈⠻⢟⣿⡉⠉⠙⢳
⠀⠀⠀⠀⠀⠀⠄⠠⣰⠈⠇⡴⠠⡀⠘⣯⢶⠀⣸⣿⣿⠃⢐⠜⠓⠀⣿⣿⣶⣶⠟
⠀⠀⠀⠀⠀⡀⣨⠡⠀⠴⠀⡄⢆⣿⣾⣿⠆⠘⠛⠋⠁⠀⠠⠂⠀⠈⣿⣿⡟⠁
⠀⠀⠀⠀⠐⠂⠀⠐⠲⠿⠃⠁⠨⢝⣿⣟⣶⠀⠀⠂⠀⠀⠀⠀⠀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠠⠠⢄⠀⢀⣠⡶⢻⡿⣽⡇⠈⢥⣤⣀⡄⠴⣷⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠑⢹⣶⣤⣬⣭⣷⣷⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢸⣿⣿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣶⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠄⠀⠀⠾⠛⠀⠀⠀⢀⠄⠱
⠀⠀⠀⠀⠀⣶⣶⣤⡀⠀⠀⠀⣀⡀⠁⠀⠰⡿⠟⠀⠀⠀⠀⠶⠀⠀⠀⡀⠀⠀⣤⠀⠀⠀⠀⠁⣘⠒⠀⣧
⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠍⠀⠉⠁⠀⠀⠀⠠⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⣴⠆⠋⠀⠍⣀⣼⠟⡧
⠀⠀⠴⠒⠒⠒⠒⠒⠒⠒⠒⠤⠤⢤⣄⣀⡀⣀⠀⠀⠉⠉⠀⠴⣀⣠⣯⠤⠀⠰⠀⠀⠀⠘⠛⠛⠛⠿⠿⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠻⠿⠿⢿⣿⣿⡿⠿⠿⠟⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⢸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⢸⣿⣷⣄⣀⣀⣠⣾⠟⠻⣷⣄⣀⣀⣠⣾⣿⡇⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠙⣿⣿⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⠋⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣷⣌⠙⠛⠻⣿⣿⣿⣾⣷⣿⣿⣿⠟⠛⠋⣡⣾⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣟⠛⠛⠓⠀⠀⣿⢻⡟⣿⣿⢻⡟⣿⠀⠀⠚⠛⠛⣻⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠛⠷⣦⣀⠀⠀⠀⠈⠁⠉⠉⠈⠁⠀⠀⠀⣀⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡄⠀⠀⣉⠳⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢉⣀⡀⢰⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⣿⡆⠀⣿⣧⠀⢰⣶⡀⣶⡆⢠⣶⣾⣦⢸⣿⠛⠃⢸⡟⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣸⡇⣸⣟⣿⡄⢸⡿⣷⣿⡇⣿⡇⣴⣿⢸⣿⠿⠇⢸⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⠁⠿⠋⠻⠷⠸⠇⠘⠟⠃⠙⠿⠟⠿⠸⠿⠿⠆⠸⠟⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠃⠀⠀⣿⠁⠘⢠⠄⣴⡢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣤⣴⣄⣺⣣⣖⣷⡀⠈⠀⢴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠀⠄⠀⠈⠉⠑⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢰⣆⣠⣤⢦⡤⠀⠀⠀⢷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⢂⡀⠚⠳⢾⠋⠀⠀⠀⠀⢸⣿⠎⠁⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⠛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇⢺⡘⠇⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣀⣀⣀⣠⠐⠉⠙⣿⣷⡄⠀⠀⡞⠧⣀⠘⢦⠀⠀⠬⠚⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠁⡹⠀⠀⠈⠀⢻⣷⠀⢀⠀⠀⠈⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁⠳⠢⠆⢁⠀⢺⡟⢠⡏⠀⠀⠀⠀⣡⠋⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢠⠆⡂⠀⠐⠲⠶⣫⢠⠏⠀⣠⡴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠁⠀⢀⠀⠀⠀⠠⠄⠀⠀⡰⠋⠀⠀⠀⡠⠂⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⡔⢀⢇⠀⠀⠀⠘⢀⠎⠀⠀⠀⠀⠀⠀⠀⢈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠝⠒⠀⠈⠁⠸⠃⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠂⠁⠀⠠⢤⣾⣞⡁⠀⢀⣀⡤⠔⠀⢀⡴⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠛⠛⠛⢅⠀⠀⠀⠀⠀⠒⠋⠉⠙⠛⠛⠯⠥⠤⣶⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⡀⠀⠀⠀⠀⠁⢠⠟⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠄⠀⠀⠀⠀⠀⢀⠘⠋⣉⡝⠁⠤⣤⡤⡄⢤⠖⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠂⠈⠙⠐⠊⠁⠀⠀⠀⠀⣀⠴⠀⣤⠖⠃⢰⣴⣶⣤⠴⡾⣷⠒⠈⣭⣤⠂⠀⠀⢀⠤⠊
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠉⣀⡠⠶⣿⠉⠉⣹⠛⠙⣿⠦⣀⣉⣳⠀⠀⠉⠁⠀⠀⠈
⠀⠀⠀⠈⡀⠀⠀⠀⠀⢀⡤⠤⠚⠉⠉⠉⠉⠀⢀⠈⠁⠀⠀⢸⠳⠀⣻⠦⣼⣇⠀⠈⢉⡟⠛⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⢨⣿⣧⠀⣿⣤⣈⡏⠁⠀⣼⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠂⠀⠀⠀⠀⠾⠿⠿⠀⠈⠛⠻⠿⠶⠚⠙⠓⠀⠈⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠮⡄⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⢧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃⠀⠀⠈⠀⢀⠠⠴⠴⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠿⢧⠀⢀⣀⠄⣴⢛⣟⠟⣻⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣈⣿⡏⡶⢻⣼⣦⡻⠿⢟⡿⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⢸⣿⣋⣹⣯⣶⠽⣿⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡓⡶⠋⣺⠍⠉⠭⣝⠿⣿⠟⠀⢀⣤⡂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣥⣾⠎⣠⣾⣿⡜⠋⣸⣏⣺⡒⠈⢢⡿⢐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣋⣼⠉⠈⠟⠉⠀⠀⠀⠙⠛⠋⠀⠀⠀⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⡿⢿⣔⠆⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡙⢬⢁⣟⠙⣻⣥⡶⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢩⣲⣦⣹⣧⣾⣌⣡⠞⠉⠉⢻⡿
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⡀⢻⢹⡀⠀⠙⣿⠁⠈⡟⠀⠀⠀⢸⡧⠴⢮⡉⠤
⠀⠀⢀⣴⠞⠋⠉⠉⠑⡄⠀⡛⢢⡴⣿⢿⡷⠀⠀⠿⣶⣾⠇⠀⠀⠀⠀⠀⠰⡾⢇⠂
⠀⠀⠙⠃⠀⠀⠀⠀⠀⠁⠈⡡⢈⠓⠉⠀⠀⠀⠘⢦⡀⡼⠀⠀⠀⠀⠀⠀⠀⠖⠈⠁
⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⠛⠄⢴⠃
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣴⠃
⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃
⠀⠀⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠟⠁
⠀⠀⠀⠈⠻⣿⣿⣶⠒⡶⢒⠀⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠂⠓⠂⠃⠈

⠀⠀⠀⢀⣀⣀⣠⢤⡤⢤⠤⡤⢦⠶⡤⢰⢦⢶⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄
⠀⢀⣰⣿⣿⣶⣷⡾⣾⣿⣷⣿⣿⣿⣿⣿⡟⢿⡿⠟⠷⠀⠀⠀⢠⣶⣿⣿⣿⠛⠒⠀⠀⢀⠀⠀⠀⠑⣄⠀⠀⠀⡰⠁⠀⢀⡴⣊⣉⠀⠸⠿⠿⠿⣿
⠹⠿⠖⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠃⠀⠀⢸⣿⣿⠉⠁⠀⠀⠀⠀⠢⠉⠀⠀⠀⠈⠂⣠⠔⢀⡤⠊⠘⠀⠁⠁⠀⠀⣤⣀⠙⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠁⠀⠰⣤⣿⣷⠇⠀⠀⠀⠄⠀⠀⠀⠀⠙⠛⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣶⡄⠀⡀⠀⠀⠀⠰⠿⣷⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡈⣛⣋⡤⠄⠀⣀⠀⠀⢛⣽⣾⣿⣮⠉⠀⠀⠤⡀⠒⢶⣶⢶⡋⠁
⠀⠀⠀⠠⢴⣄⣀⠀⠀⠀⠀⠀⠹⣬⣣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠉⠛⠧⢠⠈⠁⠀⠠⠞⡿⣷⣿⠗⠟⠆⠀⠀⠀⠀⠈⠋⠀⢳⡀
⠀⠀⠀⠀⠀⠉⠛⠻⢶⣦⣤⣤⣄⣈⣻⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠠⠀⢀⡠⠈⠀⠀⠁⢿⣿⠀⠀⠀⠀⠑⠢⠀⠁⠀⠀⠨⡇
⠀⠀⠀⠀⢀⣀⣤⣤⡔⣢⣤⣈⣉⣉⣛⡻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠏⢂⠀⠀⠀⠀⠀⠀⠀⠄⢀⡼⠃
⠀⠀⠀⠲⣿⣿⣿⣿⢿⣿⣿⣿⣿⠿⠋⠁⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⡤⠃⠀⠀⠈⢦⣀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠰⠐⠈⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠴⠄
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠷⣤⣴⡦⢉⡲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠓⣬⣛⣭⡿⢿⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣏⣷⣤⣠⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⣿⡏⡯⠘⣿⣿⠂⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⡙⡄⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣝⣦⡄⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⡳⣆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡜⢦⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠘⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡭⠼⠀⣀⣤⣂⢀⡀
⠀⠀⠀⠀⠤⠔⣒⣶⠖⣒⣤⢤⣀⡀⠀⠀⠀⢀⡴⣡⠁⣼⣗⣿⡯⣯⡷⣦⣀
⠀⠀⠐⢀⠠⢡⠬⠐⢥⡺⣏⢛⢇⣏⣋⠻⠿⣻⠿⢥⣾⠿⣟⢻⣯⣒⡾⢂⡹⠀⠀⠀⣀⣰⠛⠉⠉
⠀⠀⠀⠈⡰⠎⠼⡶⣶⣿⣬⣏⣳⣦⣵⣜⢇⡪⠠⢁⡐⡻⢝⣯⣿⡻⠫⠷⣿⣶⣶⠷⠛⠁
⠀⠀⠀⠀⠐⠄⢙⢛⣻⣿⣿⣿⠿⠿⠛⠛⠛⠛⠯⠼⠧⠭⠎⠛⠛⠁

⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢀⠀⠀⠈
⡈⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⢠⡡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣧⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣆⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⢮⣛⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣌⠳⣻⡵⡃⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠐
⠀⠦⡆⠘⢄⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠈⠯⢳⡵⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢰⡄⠙⠠⠂⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢰⠀⠀⠁⠘⢃⠉⣀⡀⠛⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⣨⠃⠀⢲⣁⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠁⡴⠂⠈⠈⣌⠁⢋⣠⠆⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⢐⠀⠱⠀⠈⢑⠀⠈⠉⠀⠀⠠⠖⡄⠀⠞⠀⠀⠀⠀⠈⣄⠩⠖⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠾
⠠⠀⡀⠀⠀⣨⠆⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠊

⠀⠀⠀⠀⠀⠀⣤⣤⣶⠾⠉⠙⠒⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣤⣴⣤⣹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⢿⣿⣿⣧⠀⢰⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣿⣿⣿⣧⣤⡹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡀⢹⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢠⣿⣿⣿⡇⢻⣿⡜⢿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⢨⣽⣿⣿⣶⣶⣾⣿⢿⣿⢱⡌⣿⣿⣌⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠁⠸⣿⣿⡇⠈⣿⣿⣦⡙⢷⡄⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣱⣇⠹⢹⣿⣿⣾⣿⣷⣄⠀⣿⣿⣿⣦⢠⣶⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠀⣌⢻⣿⣿⣿⣿⣿⣷⣿⣿⣯⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣔⣝⠶⣝⡿⣿⣿⣭⣛⡿⠿⣿⣯⣭⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⡿⣅⣨⣭⡝⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⡿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢋⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⣿⣧⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣯⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⡿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣠⡦⠄

⠀⠀⠀⣀⣀
⠀⠀⢠⣿⣿⣷⢦⣄⡀
⠀⠀⠀⣿⡟⠻⢷⣬⡙⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⣿⡇⠀⠀⠈⠙⠷⣯⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠈⠻⣿⣷⣄⡀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠹⣿⡙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⡄⠈⠻⣿⣿⡆
⠀⠀⠀⢹⡇⠀⠀⠀⠀⢰⣄⠀⢸⣷⡀⢘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣸⡇⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⢿⡄⠀⠀⠀⢀⣿⣆⠈⢿⣧⠀⠻⣿⣿⣷⣤⣀⣀⣀⣀⣤⣸⣿⣠⡟⠁⠀⢳⡀⠀⢹⣿
⠀⠀⠀⠀⠀⢻⣦⣄⢠⣼⣿⣿⡄⢈⣿⣆⠀⢹⣿⣿⣏⠻⣿⣿⠿⢿⣿⣿⣿⠿⠿⠷⣾⣷⣤⣀⣿⡀
⠀⠀⠀⠀⠀⠀⢯⡻⢿⣿⣿⣿⣿⣿⣿⢏⠀⠈⠀⠙⠿⢷⡈⢻⣶⡿⠶⡷⠶⢦⣴⣶⣤⣄⣀⢉⡉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠹⢤⡈⠙⠛⠿⢿⣿⣄⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠈⠉⠛⠛⢦⣠⡄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣓⣶⠶⠿⠿⠿⢛⣷⣶⠖⠀⠀⠀⠀⠀⣸⡟⠋⠙⠋⠉⠂⠈⠲⣤⣄⣀⠀⠉⠛⢦⣀⠙⠿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠴⢾⣿⣥⡄⠀⠒⢛⣿⣿⡏⠀⠀⠀⠀⠀⠀⢤⣽⣿⢠⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄⠀⢻⣧⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⣠⣶⡿⠟⠋⠀⠀⠀⠀⠀⠀⣀⣀⠈⢉⡀⢿⣶⣄⡀⢠⡀⠀⠀⠀⠈⢻⣿⣆⠀⠛⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡧⠀⠾⢻⡟⠁⠀⠀⠀⢀⣴⣿⠃⠒⠺⢿⣿⣿⡿⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠐⣿⡟⡄
⠀⠀⠀⠀⠀⠀⠋⣿⠟⠁⣠⣾⣿⠋⠀⣠⠄⠀⣼⣿⡋⠀⠀⠀⢀⡈⠀⠀⠀⢻⣿⣿⠉⠛⢿⣷⣄⠀⠀⠈⢹⣧⠀⠀⠀⠀⠀⢻⠀⢰
⠀⠀⠀⠀⠀⢀⣼⠏⠀⡰⢫⡿⠁⣠⣾⡏⠀⢸⣿⣿⡇⠀⠀⠀⠘⣿⡆⠀⠀⠘⣿⣿⡄⢀⣈⣿⣿⣷⣦⣀⣼⣧⣤⣄⣀⣠⡆⢰⠀⣾
⠀⠀⠀⠀⠀⢸⣿⠇⠀⢠⡿⠀⠘⣫⣿⡇⠀⢸⡛⣿⣧⠀⠀⠀⠀⠙⠃⠀⠀⠀⠘⢿⣿⣾⣿⣿⣿⣿⠋⠙⠻⣿⠄⠀⠉⠉⢳⣿⣤⡆
⠀⠀⠀⠀⠀⣿⡟⠀⢀⣿⠁⠀⠐⣿⣿⡀⠀⠘⠃⠙⠻⣿⣦⣤⣤⣀⣀⠉⠙⠓⠶⠾⠿⠿⠿⠿⠿⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠻⣃⡇
⠀⠀⠀⠀⠀⠋⠁⠀⢸⣿⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠠⠄⠚⠛⠛⣛⠒⠒⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⡀⠠⣄⡀⠠⣀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⣠⡄⢸⣿⠀⠀⠛⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣦⣤⡀⠀⢠⠈⠹⠟⠳⠄⠀⠉⠳⣤⣈⠙⠦⠈⠳⠆⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠘⢻⣇⢸⣿⡆⠀⢀⣿⣧⡖⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⡿⣿⣷⡀⢧⠀⠀⡀⠀⠀⠀⡀⠈⠻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢸⣿⡎⣿⣷⠀⠸⣿⡟⠀⣰⢃⣾⣿⣿⣿⣿⣿⣿⣏⠙⢿⡎⠙⣿⣿⡇⢸⠁⠀⣀⠀⠻⣆⠀⠟⠀⢰⣶⣦⣄⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠈⠙⣇⢹⣿⡀⠀⣿⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡆⠈⣿⠀⠈⠙⣿⡐⣦⠀⠙⢦⠀⠙⠀⠀⠀⠘⠛⠻⣿⣿⣷⣶⡶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⣿⡇⠀⠋⠀⣿⣿⣿⣿⣿⠿⠋⠙⣿⠁⣿⠇⠀⠈⠳⡀⠀⠛⣷⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠀⠈⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⠃⠀⠀⠰⠃⢀⠟⠀⠀⠀⠀⠣⢄⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⢸⣿⣿⣿⡧⠖⢨⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⠛⠛⠻⣦⣄⡀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢪⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠘⣿⡟⠁⠀⠀⣼⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⡀⠐⠋⠉⠛⠿⣷⣶⣶⣾⣿⣧⣄⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢰⡄⠀⢿⣦⠀⠀⢰⣿⣆⢈⡄⠀⢰⡄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢉⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢿⡀⠘⣿⣧⣤⡟⠀⠉⠉⡹⢁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠘⢷⡀⠈⠻⠿⣿⣷⣶⣾⡤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⡀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠋⠁

⣿⣿⣿⠿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠉⠉⠁⠉⠉⠉⠙⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢰⣏⣤⣶⣶⣶⣶⣤⡀⠀⠀⣠⣴⣶⣶⣶⣶⣄⢻⡄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣤⡜⢿⣿⣿⣿⣿⣿⣿⣿⢻⣮⡇
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠘⢿⣝⢿⣿⣿⣿⣿⠿⠋⣼⣿⣿⡌⠛⠿⣿⣿⣿⣿⠟⣸⡿⠃⠀⠀⢀⣠⠤⠶⠦⣄
⠀⠀⠀⢀⣠⣼⡅⠀⠈⠙⢧⡄⠀⠀⢈⣿⣷⣦⣭⡀⠀⠀⢸⣿⡿⣿⡷⠀⠀⠀⢀⣤⣴⣿⣿⠁⠀⢀⡴⠋⠀⠀⠀⣤⣟
⠀⠀⠀⢻⡅⠀⠀⠀⠀⠀⠀⠙⠳⢦⣼⣿⣿⣿⣿⡇⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⡶⠞⠋⠁⠀⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠀⠉⠳⠦⠤⠤⠤⣤⣀⠀⠀⢸⣿⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⠃⢀⣀⡤⠶⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣾⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡶⢤⣀⡈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢀⣀⣤⣾⡋
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠴⠚⠉⠁⠀⠀⣈⣩⠿⠓⠒⠒⠒⠒⠒⠒⠿⢯⣉⠁⠀⠀⠉⠙⠲⢤⣄⡀
⠀⠀⠀⠀⣠⠞⠋⠉⠉⠉⠉⠀⠀⣀⣠⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠦⣤⣀⠀⠀⠈⠙⠛⠉⠙⠛⠶⣄
⠀⠀⠀⠀⠻⣤⡄⠀⠀⠀⢠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⡄⠀⠀⠀⠀⣀⣀⣸⠇
⠀⠀⠀⠀⠸⣇⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣼⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉








⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠛⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣷⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠶⠶⢤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡇⠀⠀⢀⣠⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢾⣿⣿⣦⣶⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣅⡀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢉⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣴⣶⣿⣴⡚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣟⠛⠻⠿⠛⠛⠻⡿⢭⠙⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣤⣀⣀⠀⣠⢆⣀⣀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⡗⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⣧⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠐⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉
⠀⠀⠀⠀⠀⢀⡴⣫⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⡀
⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣽⣯⡷⣄
⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣉⢿⡧⠑⢤⠤⢀⡐⠒⠖⢂⢤⢀⣀
⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⢿⠃⠀⠀⠀⠀⠀⢀⢠⠤⡒⣘⢊⡽⠦⠤⠄⠑⠚⢫⣢⡄⠀⠐⢻⠻⠿⣎⡛⡳⡀
⠀⠀⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠙⣴⡂⢀⡠⠐⠅⡐⠐⠑⣈⣨⣤⣴⠲⠆⠀⠀⠀⠠⢭⣄⣤⡺⢷⢾⡶⠿⢿⣿⢿⣷
⠀⠈⣿⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⠜⠋⠁⠀⠐⣁⠤⢚⡩⠔⠚⢁⢀⠄⢒⠀⠀⣤⠀⡷⠀⠀⠘⠁⠛⠚⠛⠓⠉⠉⠁
⠀⠀⢿⣄⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠁⠄⠀⠀⠼⠛⠋⠁⡀⠀⠀⠀⠔⠑⠁⠀⠀⢸⡀⢱⣄⠀⡀
⠀⠀⠈⢿⢿⣾⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠙⠁⠀⠀⠐⠐⠤⣐⢠⡪⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣎⣩⡷⢯⣀
⠀⠀⠀⠈⠻⣿⣿⣾⣕⡲⠶⠶⠶⣒⣾⣫⣢⣥⣤⣢⡠⣠⣤⡒⣡⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠈⢣
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣝⣿⣭⢿⣿⣿⣿⡿⠟⠛⠋⢁⡼⢉⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⡎⣩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣮⡶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣟⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣳⣿⣿⣿⣿⢿⣿⣿⢿⣝⢿⣿⢿⣿⣿⣷⣻⣿⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣞⣻⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⣿⣿⣿⣿⣟⣿⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣾⣿⢿⡿⠛⠉⠉⢹⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣾⣿⣟⢿⡟⠈⠟⠻⢿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⢻⣿⣿⣻⣿⣿⣿⡘⢻⣿⡿⠁⠀⠉⠙⠳⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⣿⠿⠟⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣷⣄⡁⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⣿⣾⠿⣿⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣯⣿⣿⣿⣿⣿⣤⣀⠀⠀⠸⢿⣿⣿⡿⣂⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣽⣿⢿⣿⣿⣟⣿⣿⡻⣾⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⢿⣿⣿⣻⣿⡤⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣤⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⣿⣿⣿⣿⡿⠛⠛⠛⠻⣿⣿⣿⣿⡿⠛⠛⠛⠛⢿⣿⣿⣿⡿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⢀⣀⣴⣿⠟⡟⢿⣷⣄⣀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡏⠀⣷⠀⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣯⠙⣿⣿⣷⣼⣿⣴⣿⣿⣿⠩⣿⣏⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠋⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⠛⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⣤⣤⠄⣀⣠⣤⠴⣶⣶⠶⡒⠒⠒⠶⠆⠀⣀⣤⣴⣶⣿⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡷⢖⠛⡩⢶⠃⠿⢟⣫⣽⣶⣶⡜⢟⡴⣛⠽⠻⣟⣭⢾⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⢏⣴⠇⣠⡴⢖⣒⣛⠉⢷⡁⠀⣹⠗⡞⠀⠁⠠⣾⣿⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⢻⣿⡾⢸⢏⡶⢋⣽⣧⠶⠿⢃⡤⢂⣤⣦⣄⠀⢀⣶⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠈⡏⡼⠋⠘⣠⣴⣯⣵⣦⣤⣬⡽⠞⠫⠵⢦⣌⣧⠸⣿⢡⣟⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡷⣰⠃⠀⣸⣿⢻⣤⡥⣋⠕⡊⡲⣶⣦⡴⢣⡿⠀⠀⠃⣸⣿⣆⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⠄⠁⠀⢰⠋⠀⢀⣤⣬⣤⠄⣽⣇⡴⢂⠀⣯⣜⠏⡀⠀⣿⣿⣹⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⡇⠀⠀⠀⠸⢦⣤⠄⠀⠀⢒⣭⢿⣯⡶⣻⠏⠋⣙⣴⠃⠠⢿⣿⠉⡟⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣗⣻⡿⠃⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣶⠿⠟⢁⠄⠀⠸⣿⣾⠃⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠙⢿⡁⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣲⣶⠶⠞⢉⡀⠂⠆⢻⡏⡇⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⠳⣤⣀⣙⣿⣿⡿⣿⣿⣿⣿⡿⢿⣿⡿⣾⢁⡀⣴⣃⣾⠀⣾⣿⢸⡿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣤⣼⣷⢾⠋⠀⠈⣇⣿⡆⢻⡇⣿⢋⡼⣻⣫⡏⠀⣿⡟⣾⠃⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢏⠁⠙⠅⠈⠀⠀⢀⣸⢇⡏⣸⣿⣷⠏⣸⣩⠟⢀⣾⣟⣽⣫⣦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣼⢡⣟⡾⠁⢠⡟⣡⣼⠿⣫⣾⢁⣥⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⡼⣡⠟⣋⣤⣴⠿⣛⣩⣴⠟⠫⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⣠⠟⣛⡵⣟⣯⣷⠾⠟⢛⣉⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⡡⠂⣴⢃⡾⣫⠾⠛⡉⠀⢀⣶⢛⡍⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢠⢺⢇⡞⡽⢁⡞⠋⠀⢸⠟⣙⠋⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡌⢸⢸⠣⠇⠈⢧⣀⣀⣾⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠁⢸⢸⡄⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠘⣆⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠈⠆

⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⠀⠤⠀⢀⣄⠸⠀⠀⠀⣤⣶⡄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡘⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠻⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢰⡖⢰⣶⢄
⠀⠀⡼⠁⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡽⠟⠛⠘⡅⢆⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤
⢀⡌⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠐⡀⠀⠀⠀⠀⠐⠆⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠇⠀⠀⠀⠀⠀⠦⣄⠀⠄
⣿⠇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⢤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠤⠄
⠋⠀⠘⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⢀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢄⠀⠠⠀⡀⠀⠑⠩⣢⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡔⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀
⠈⠀⠀⣦⡈⡢⠄⡀⠀⠀⠀⠀⠀⢉⣉⣙⡛⠙⠋⠀⠀⠀⠀⠐⠠⢀⠀⡠⡀⠰⡃⠀⠠⠎⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠂⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢶⢶⣦⣃⡀⠀⣤⣀⣉⣩⣭⣍⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠲⣆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹
⠀⠀⠀⡃⠸⣿⢿⣿⣿⣿⣿⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠁⠘⣄⠀⠄⠀⢀⡀⠀⠀⠀⢀⠀⠂⠁⠠⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⢀⠀⠀⠘⠛⢿⣏⣛⠙⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢈⡁⠀⢀⡀⠈⠧⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⢤⢀⠀⠀⠀⠀⠀⠈⠙⠳⠀⠀⠀⠀⠐⠀⠀⠀⠀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢐⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁
⠺⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠰⠂
⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⣦⡀⣿⣄⡄
⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠘⣣⠈⠓⠘⡦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⢰⠀⠀⠀⠀⠢⠀⠐⢄⠐⠀⠀⠀⠘⠋⠁⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠄⠘⡀⣄⢀⢃⠀⠀⠀⠀⠁⡞⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⢠⠼⠏⡐⡶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠈⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⡈⠀⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠰⣟⣿⣿⣿⡿⠟⡻⠶⠖⠲⢆⣀⣀⠀⠀⠀⢃⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠔⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⣠⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠃
⠀⠀⠀⠀⠀⠇⠀⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⣄⠀⠀⠀⠀⠀⠃⢠
⠀⠀⠂⢄⣀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄⠘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉
⠀⠀⠀⠀⠀⠙⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠂⠀⠀⢀⠔⣩⠖⠈⠉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢠⣶⠠⠀⠀⠀⠀⠀⠁⠠⢁⠎⠀⢀⠔⡀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⢋⠞⡴⠁⢀
⠉⠀⠀⠀⠀⣘⠿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡌⠐⠀⢠⠃⡠⢀
⠀⠁⠀⠀⠀⠉⠲⠠⠄⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠂⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⢁⠂
⠀⠀⠀⠀⠀⢀⡄⣧⠀⠀⠁⠄⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⢸⠇⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣘⡰⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠹⢬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢷⡠⠶⣮⣔⣀⠀⠀⠀⠀⠀⠀⠧⣄⡉⠲⢤⡀⠀⡠⠖⢒⡚⠛⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠂⠠⡄⠈⡇⠀⠀⠀⠀⢀⣀⡤⠽⠒⣤⠙⢲⢲⠞⢉⣀⡀⠀⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⠁⢀⡇⢀⡠⢔⠊⠑⠀⣿⠀⠰⣿⡀⠸⠆⠀⠙⠿⢋⣨⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⡸⠟⠛⠀⣾⣦⠀⠐⡛⠲⢀⠈⠉⢀⠟⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠐⠉⠙⣶⠆⠀⠙⠿⡥⢈⣡⠊⢁⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣖⡋⢛⡆⠀⠘⠃⢀⣋⡤⠴⠚⠉⠣⣄⡙⢄⡀⠀⣴⠆
⠀⠀⠀⠀⠀⢀⣠⣴⡚⢿⠅⣙⣇⠨⠥⠚⢄⠈⠈⡏⠉⠀⠀⠀⠀⠀⠀⠀⠉⠒⠬⡉⢳⡚⠁
⠀⠀⠀⢀⡴⠿⣄⡬⠗⠊⠉⠀⠀⠀⠀⠀⠈⢢⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⣏⠙⠿⠂
⠀⠀⢀⠞⣢⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋
⠀⠀⡞⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⡤⠊
⠀⢸⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠔⡱⠋
⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠜⢢⠺⡲⣲
⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠛⠂
⠀⠀⢳⠹⡀
⠀⠀⠀⠳⣕⠤⣀⣀⣠⠔
⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠁⠀⠻⠆⠀⣠⣤⣾⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣶⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠿⢛⣩⣵⣶⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣰⣿⣿⠟⢉⣤⣾⣿⣿⣿⣿⣿⣿⠟⠉⢀⣄⠀⠀⠀⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⠋⣠⣿⣿⣿⣿⠟⣿⣿⣿⣁⠀⣀⣾⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⢰⣿⣿⣿⠟⠁⠀⠛⠿⠿⠿⠿⣿⡿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢻⣏⣿⣿⠀⣿⣿⣟⣥⣤⣄⠀⠀⠀⠀⠀⢰⡇⠹⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣄⡙⢷⣬⣿⣿⣿⠀⣿⣿⣿⣿⣿⡟⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣉⣹⣿⣿⡿⢹⣿⠀⣿⠟⠋⣿⣿⣤⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠋⣵⠏⠀⠀⠸⣿⡄⣿⠃⠀⠉⢉⠿⣏⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⠀⠀⠀⠛⠀⠙⠁
⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣴⣿⡿⠿⣿⣿⣦⠀⠀⢠⣿⣿⡟
⠀⠀⠀⢰⣿⠋⠀⠀⠈⠻⣿⣷⣶⣿⣿⡿⠁
⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠉⠛⠿⠛⠋
⠀⡀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀⠀⣀⣀⣤⣄⣄⣀⣀⣀
⠀⠀⠀⢀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠋⢉⣉⡀⢀⣈⣉⣉⠛⠛⣉⣤⣤⠀⢩⣍⣙⠛⠶⣦⣄
⠀⠀⠀⣾⠃⢀⣀⡈⠉⠛⠛⠶⢶⣦⡴⠞⠋⠁⠀⢀⡖⠋⠉⠀⣴⠏⣉⠉⠉⣽⣛⣋⣴⠖⡲⠀⠀⢹⡇⠀⠀⠙⢿⣦⣀⣤⣴⠶⠶⠟⠛⣷⡄
⠀⠀⠀⢿⠀⠈⠉⠻⢿⣶⣤⣀⡀⠀⢠⣀⠀⠀⠀⠊⢀⣠⣶⠟⠀⣿⡁⠀⠀⣼⠀⠘⠃⠞⠇⠀⢀⡌⠳⢦⣴⣇⣀⠙⠏⣀⣤⣶⣿⡷⠂⣿⠁
⠀⠀⠀⠸⡇⠀⠀⢰⣿⣿⣿⣿⡏⠀⠠⣬⠳⣄⠀⠙⠛⣩⡿⠷⢿⡟⠃⠀⣈⣵⣀⣀⢀⣐⣶⣾⣋⣭⡙⣾⡷⣄⠸⡆⢀⡉⡻⠟⠉⠀⣸⡏
⠀⠀⠀⠀⢻⡀⠀⠀⣙⣿⡿⠁⠀⠀⣀⠩⣤⣌⠳⡝⣿⣏⣼⣿⣶⣿⡟⠛⣫⡽⠿⠭⠿⠶⠶⢿⡟⠻⠟⢛⣿⢈⡣⠶⣏⠤⠈⠃⠀⣴⡟
⠀⠀⠀⠀⠀⠙⣦⡀⢉⣿⠍⠉⢛⣒⡭⢤⣀⠙⢷⣝⣮⠻⠿⢿⣶⣤⠁⠘⢁⣠⣤⣤⣤⣤⣤⣤⣉⠃⠐⠫⢖⣩⣴⢺⡅⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⣸⠷⣾⣯⠥⢤⣘⣛⠃⠐⠾⣿⠶⢭⣝⡓⠀⠴⢤⡀⠀⠀⢻⣿⣭⣤⣤⣤⣤⣼⡟⠀⠀⠪⢭⣤⣖⣲⣶⢒⡂⢺⣷⡀
⠀⠀⠀⠀⠀⢠⠏⣞⠟⢻⡧⣤⣙⣛⣉⣻⠷⠒⠒⠦⢬⣽⣓⠂⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⠟⠉⠀⠀⠀⠀⢁⣉⣀⠡⢽⡭⠁⢺⣿⠃
⠀⠀⠀⠀⠀⣿⡈⠐⠀⣸⡧⢬⣭⣥⠤⢽⣯⠭⠭⢍⣛⣒⡦⠀⠀⠀⠀⠀⣀⣀⣠⣤⣼⠿⢶⣦⣤⣤⡀⠀⣐⠭⣈⠉⢹⣇⠀⠺⣿⡆
⠀⠀⠀⠀⢠⣯⠔⡢⡀⣸⣿⡀⢻⣿⡀⠘⣿⠀⠀⠄⠶⣖⣻⡅⢀⣤⣶⣋⣉⣁⣠⣤⣤⣤⣤⣀⡤⢍⣻⡾⠋⣑⣈⠁⢸⣇⠀⠙⣮⣷
⠀⠀⠀⠀⣼⠡⡪⢞⠁⣼⣿⡆⠸⣿⡇⠀⠻⣦⣄⠀⠉⢻⣿⣿⣿⣿⠀⠀⣿⣄⣧⣦⣦⣤⣤⣼⠀⠀⣿⠇⠀⣹⠃⠀⣿⡗⠀⠀⠩⣻⣧
⠀⠀⠀⠀⣿⢠⢖⡡⠀⣾⣿⠃⢐⣿⣿⣦⡀⠘⢧⠀⠀⠈⢿⣿⠯⣿⡄⠀⡟⡿⡟⡟⣿⢻⣿⢿⡀⢸⣧⣤⠾⠃⠀⢀⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢿⡶⣫⠖⢀⣿⡇⠀⠈⠸⣿⣿⣷⡀⠈⡇⠀⠀⠼⣿⣢⠙⣿⣄⣿⠉⠧⡇⠟⠈⠸⣸⣷⡏⣼⠁⠀⢀⣦⣿⠿⠃⠀⡀⠀⣜⣿⡇
⠀⠀⠀⠀⢸⡆⠁⠀⢠⣿⡆⣸⡟⡀⠉⢻⣿⡇⠀⢻⡆⠀⢸⣿⣿⣁⢻⣿⣿⠀⠀⠀⠀⠀⢠⢿⡏⣷⠿⢿⣿⣿⡏⠈⠀⠀⢱⣱⣦⣼⣿⠃
⠀⠀⠀⠀⠈⣿⣴⡄⡄⠻⡿⣿⣿⡅⠀⠘⠛⣗⠀⠈⢳⣄⠸⣿⣿⣯⣾⡇⠸⣦⣤⣤⣤⣴⠏⢸⣇⣿⠟⠀⠀⠈⠀⠀⠀⣠⣣⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠹⣿⣧⣷⢠⢠⠡⢻⣦⠀⠀⠀⠀⠀⠀⣈⠛⣶⣾⣿⣿⠏⢧⣀⣉⣥⣤⣭⣏⣀⣸⢻⣿⡆⠀⠀⠀⢠⣀⣰⡟⢳⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢾⣷⣤⣿⣬⢠⢰⠀⢠⢃⢅⣧⣉⢹⡵⠛⣯⣷⣶⠶⠶⣦⣤⣤⡤⣬⣠⢟⡀⢦⢦⢰⣸⢻⣿⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⣧⣦⣧⣮⡿⠿⠿⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣼⢸⡞⡄⢾⣸⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⠋⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡟⠻⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⢠⡀⣄⠀⠀⠀⠀⢀⣠⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣦⣼⣶⣴⣾⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢄
⠀⠀⣀⣀⣀⣠⠴⡶⠐⢎⣲⠦⠔⠊⠉⠉⠉⠉⠉⠓⠒⠶⠿⢟⡁⢂⡀⠀⠑⡄
⣦⠞⢁⡉⠨⠇⢀⠀⠊⠙⠗⠊⠉⠁⠂⠀⠀⠀⢀⡴⢐⣿⣿⣷⣮⣿⣝⣶⡀⠙⢆
⡇⡄⠊⢡⡶⠂⣃⠀⠀⡀⠂⠤⠉⣑⣈⠁⠒⢿⠟⣏⠘⢏⡻⠻⠛⠝⡳⢽⡍⠁⠘⢇⡀
⣧⣷⠶⢻⠄⡠⠈⠀⠀⠈⠠⠤⠄⠐⠁⠈⠉⠁⠀⠊⣱⣆⣘⠿⠛⢁⣠⠤⠂⠂⠀⠘⠻⣄
⣧⡼⢅⡼⢁⡀⠀⠠⠐⠛⠛⠒⠒⠶⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠉⠁⠀⠀⠈⠓⠬⠶⠄⠤⣀⠀⠀⠀⠀⠀⢀⡤⣄⣀⡀
⡇⣀⢌⣷⠸⢷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠙⠠⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠴⠛⠊⣰⠖⠊⠛⠶⠍⠙⢷
⡇⡼⠐⢻⠷⡄⢿⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⠦⠄⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣇⣹⠲⢶⢼⣉⣪⠂⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠈⠁⠎⠵⠉⡼⡿⠲⢤⡀⡐⠂⠀⠀⣀⢀⢰⠶⢄⡀⠀⠀⠐⢠⢤
⡗⢴⡆⠴⠴⢇⣀⣷⣄⡀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠟⢀⠞⠀⣬⠉⠀⠀⠃⠰⠃⢢⠍⠐⣘⣫⠃
⡧⠼⣀⡀⣷⠊⠹⣄⣹⡯⣷⡒⢶⠦⡤⢤⣤⣀⣀⣀⣠⣤⠤⠤⠒⠲⠶⠶⠒⠒⠒⠒⠒⠒⠚⠒⠒⠒⠀⠉⠉⠛⠛⠛⠋⠉⠉
⡇⡴⠛⡤⠼⢠⠖⢢⡽⢦⡞⠉⠿⢬⡧⢴⣁⠴⠚⠛⠉
⠛⠓⠒⠛⠒⠚⠓⠚⠓⠚⠛⠒⠚⠓⠛⠋⠁

⡀⠀⢹⡄⠀⠀⠀⢀⡀⡂⠀⠀⠀⣠⡄⠊⠀⠀⠀⡀⠈⢿⣇⠀⠃⢀⣀⠀⠛⠻⣐⠿⢤⠔⠒⠤⠄⣿⡾⠟⠀⢊⣿⣿⣿⠏⠈⣯⡐⢀⠀⠀⠀⠈⠒⠤⣤
⠁⠀⠀⠙⡄⠀⠀⣤⡀⠡⠶⢶⣼⣯⠕⠀⠀⢂⠀⠀⡀⣸⡿⠀⠘⠀⢩⣄⠀⠀⠎⣤⢤⡴⠶⠟⠘⠈⠀⠀⢀⣌⢶⡟⡻⣄⠀⠋⠐⢦⣣⣄⡀⠀⠀⢀⣀
⡀⠀⢀⠀⠙⠂⠒⢿⡈⠳⢤⡈⠛⠻⠀⡀⠠⡤⠀⠀⠀⡿⠃⠐⠀⠀⢾⠋⡥⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠊⠁⠈⠋⠑⠫⡀⠀⠀⠈⠉⠻⠻⠆⠉⠉
⡇⠀⠸⣆⠀⠀⡀⢈⣷⡀⠀⠙⠣⢀⠈⠈⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⣇⢀⠀⠘⡆⠀⣇⠀⢿⠛⠢⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠟⡌⠀⠀⠋⠂⠄⠀⢠⡀⠀⠈⠓⢆⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠠
⠀⡀⠀⠀⠠⠀⠀⠀⠀⠈⠒⠄⡀⠀⠉⠓⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⣘⠀⠀⠀⢀⠀⢂⡐
⠟⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠠⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢻⡄⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣿⣦⣤⣀⠉⢷
⠀⠀⠀⠀⠃⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠈⣉⣙⠛⠶⢬
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⢠⡤
⢆⠀⠀⠀⣶⣠⣥⡀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣬⡀
⡛⣷⠀⠀⣮⣽⣷⠞⠁⡀⠀⣠⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠤⠄⠀⠀⠀⢀
⡻⣾⡇⢸⣿⣯⣿⠀⣸⠗⠊⠖⠉⠉⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠘⠓⠲
⣿⠿⣿⣼⡇⣸⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢹⢶⣽⠙⣷⣿⣥⣄⣤⡴⣆⠀⢰⠴⣦⠠⢄⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠠⡀⢀⣦⠀⠀⠀⠠⠀⣴⣟⠉
⠀⣿⣽⣷⡌⠻⠟⠛⠉⠁⠀⠀⡀⢅⣨⣤⡀⠀⠀⠀⠀⢀⠃⠀⠀⠂⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣿⣃⠀⠀⠀⠀⢀⠀⠈⠻
⠂⢿⡏⠙⠟⠧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⡄⠀⠀⡜⠀⡠⠀⣼⠏⠀⠀⢀⡤⠤⠎⠀⠀⠀⠀⠀⠁⠀⢀⡅⠀⠀⠀⠀⠀⠀⢰⡖⠒⠁⠊⢄⠈⠐

⠀⠀⠀⠀⠀⠀⠠⠴⠔⠒⠒⠶⠤⣄⡀
⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢
⠀⠀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢠⣠⣀⠀⢀⣤⣤⠀⠀⠀⠀⠀⢀⣀⡠⣤⣤⣄⣠⣤⡴⠶⢤⡴⠚⢛⠉⢉⣭⣉⠉⢐⡆
⠀⠘⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢷⣿⣷⢋⣿⡿⠖⠐⠒⠚⠙⠉⠀⠀⢀⠀⠀⠁⢈⡀⠀⠀⠂⢰⣦⣾⣿⣿⣿⣿⡿⠃
⠀⠱⠊⣸⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣰⠎⠻⠛⢛⢠⢄⣠⣄⣄⠀⠀⢠⠀⠀⢀⣐⢀⣠⢈⠳⣴⣤⣳⣾⠿⣿⠛⠛⠁
⠀⠀⠘⠥⡄⡉⠹⠟⢟⠲⠺⠚⠿⠋⠉⠀⠐⠀⡄⠀⠰⣿⣿⣿⣿⣟⣿⣷⣾⣶⣶⣿⣿⣿⣿⣿⠇⢀⣴⠟⠫⡒⠺⠉⠄
⠀⠀⠀⠀⠈⠉⠙⠷⠶⢶⣤⣆⣀⣄⣀⣀⣄⣄⢴⠦⠾⠟⢻⡟⣛⣛⣯⡵⠿⠿⠿⠟⠛⠛⠉⡁⠠⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠻⢿⣿⠛⡛⠃⢀⠀⠀⠀⠀⠀⠀⠉⠐⠠⠬⠽⣒⡁⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡶⠶⠶⠶⠶⠶⣦⣤⣀
⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⠠
⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⢤⢠⣤⠘⠰⢀⡉⠁⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠠⡄⠑⢄⠀⠐⠔⠀⠐⣆
⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⣳⣆⣒⠻⣂⠽⣦⣄⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠁⠀⡈⠀⠇⢧⡰⡄⠹⡆
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢈⡻⣷⣟⣻⡴⠅⣌⠓⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣍⡳⠈⢜⢪⡃⣄⢢⣤⠄
⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⣠⣴⣾⣿⣿⡿⣿⣏⡿⠿⡿⣿⣯⣿⣿⡟⢿⣿⣿⣿⣶⣄⠀⠀⠀⠙
⠀⠀⠀⢘⣿⣿⣦⡀⠠⣤⣤⣴⡀⣀⠀⠀⣀⣴⣿⣿⣿⡿⣻⣭⣏⣲⣇⣱⠊⠆⠈⠉⡁⠞⢉⡟⡻⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠹⣿⣿⣿⣶⣤⣭⣿⣿⣤⣶⣾⣿⣿⡿⣿⣶⣻⣯⣿⣿⣿⢿⣿⡞⣄⢀⣀⡄⠀⠈⠀⠐⢺⣽⣿⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⡡⢔⠺⡞⣏⠟⢿⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⣿⣿⣛⡛⠛⢻⣿⣿⣿⣿⣿⠻⣿⣏⣿⡿⣿⣿⣿⣿⣾⣾⣦⣠⣠⣀⢀⣠⣩⣻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠠⠤⠤⢤⣽⣿⣿⣿⣿⣶⣤⠄⠀⠀⠑⠓⠈⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠉⢉⠶⠚⠉⣩⣁⣉⠙⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⡿⠿⡽⣿⣿⣷⣤⣄⢈⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠒⠚⢿⣿⣿⠿⣶⣶⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠿⣿⣿⣿⣿⡆⠁⠀⠙⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠗⠂⠀⠈⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠛⠂

⣶⣿⣷⢿⣶⠿⣷⣿⣿⢶⣶⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣶⣾⣾⣷⣶
⣿⣿⣦⣶⣵⣶⣤⣼⣶⣾⣧⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣤⡘⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣤⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⠛⣷⡈⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⣿⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⢠⣾⣿⠀⣿⣷⡌⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠒⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⠛⢐⣟⢻⣿⣭⣿⣿⣿⣆⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣭⣭⣭⣭⣍⣭⣭⣭⣭⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⠤⠦⠄⢀⠀⡀
⠀⠀⢠⠙⠀⠄⡐⢏⡡⠀⠀⠀⠈⠀⠁⠐⠀⡀
⠀⠀⠀⠉⠐⠂⠠⠀⠄⠚⠙⠚⠶⢆⡀⠀⠀⠀⠃⡀⠀⠀⠀⡀⠄⠂⠀⠒⠀⠰⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡄⠀⠄⠘⠀⡠⠊⠀⠀⠐⠀⠀⠠⠀⠀⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠀⠀⠁⡖⠀⠀⢀⡴⠊⠉⠈⠙⢣⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠄⣸⡅⠀⠀⠌⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⢀⠀⠌⢸⠤⠀⠀⠃⠀⠀⠀⠀⠀⡼⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠣⠀⡀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠬⠁⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠂⠁⢀⠀⠀⠀⠀⠈⡠⠀⠀⠈⠀⣠⠋⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠈⠀⠀⠀⠀⠀⠀⡵⠀⠐⠀⡴⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠄⠀⠀⠑⠤⠀⡤⠚⠅⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠲⠴⠐⠒⠉⠀⠀⠀⠣⣀⠀⠀⠁⠂⠠⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠓⠒⠦⠤⠤⣄⣀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⣀⣠⠔⠁⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠉⠃⠀⠀⠁

⣿⡏⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⢶⣄⢀⣀⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣼⣿⠉⢻⣇⢸⣿⡟⢙⣿⠀⠀⣾⣿⣟⡛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠸⣧⣄⠉⣉⣙⣛⣿⣿⣦⣏⣴⣿⣯⣤⣤⣾⠿⠟⠁⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣦⣤⣾⡟⠋⠉⠉⠛⠷⣶⣄⣤⣶⡿⡟⠋⠩⠭⠭⡉⠫⠍⠍⠁⠀⣀⠀⠀⣀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⠀⡄⠉⣉⠉⠉⠛⠳⠶⣷⣦⣠⣾⠟⠉⡠⠞⠀⠀⠐⠒⠊⠀⣔⡢⠀⠀⣸⣶⢿⡿⠉⣹⡷⠿⢿⣦
⠀⠀⠀⠀⠀⠀⢀⣴⠾⢛⣿⠿⠃⠀⠀⣒⠉⠀⢘⣴⠶⠶⠶⢤⣀⠠⡉⠁⠀⠀⠀⠀⡄⠒⡆⠑⠃⠀⠀⠀⠀⠀⢸⣯⣀⣼⡇⡴⢋⣠⣤⣾⡏
⠀⠀⠀⠀⠀⠀⣾⢃⣾⠿⠓⠁⡀⢀⠈⠒⠁⢠⡟⠁⡔⣆⠀⠀⠙⣆⠹⡄⠀⠐⠵⠀⠑⠈⠀⠀⠀⠀⠀⢀⣦⡀⠀⠙⢛⣉⡀⣼⣟⡋⠉⠉⣻⡇
⠀⠀⠀⠀⠀⢀⣿⠞⠁⢀⣀⠈⠄⠌⠀⠀⠀⢸⡀⠀⢃⢿⡇⡀⠀⣿⠀⠃⢀⠀⢐⣁⠆⠀⠀⠐⠒⠛⠻⣿⡟⠿⠾⠟⠛⠛⢿⡌⠻⣿⡿⠿⠛⠁
⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠙⠀⠀⢠⠒⡄⠀⠈⢷⣀⠀⠂⠕⢀⣴⠃⠀⠀⢸⡆⠀⣀⣠⣤⣤⣤⡄⠀⢰⡿⠁⠀⠀⠀⠀⠀⠸⣷⣀⣸⡷
⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠛⠓⠛⠋⠀⠀⠀⠀⣾⡿⠛⠉⠉⠁⢠⣿⠁⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁
⠀⠀⢰⣿⠀⢀⣴⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⣸⡟⣀⣀⣀⠀⣀⣾⠇⠀⣾⠏
⠀⠀⠸⣿⡀⠀⠁⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⣰⣿⠟⠛⠛⠛⢛⣛⡉⠀⠀⢿⣦
⠀⠀⠀⠙⢿⣦⣀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣤⣞⣡⡴⠟⢀⣠⣾⠟⠻⠷⣶⠶⠿⢛⣿⠇⣤⠰⣦⡙⢷⣤⡀
⠀⠀⠀⠀⠀⠈⠙⠛⠿⢶⣶⣶⣤⣴⣶⣶⠿⠟⠻⠿⣷⣾⣿⡿⠟⠁⠀⠀⠀⠀⠀⣴⡿⠋⢸⣿⣧⢻⣿⡆⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢀⣿⠏⣿⠀⢻⣿⣦⣀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⡾⠋⠀⣿⣄⣸⣿⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋⠄⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡚⣟⣳⠀⠀⠀⠀⢀⡀⢇⣀⡠⠰⣃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣻⠻⢲⠖⣠⢤⣤⡔⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣳⡭⠼⠿⢷⣀⠀⠀⢀⡽⠋⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠃⠈⠉⠁⠰⣄⠀⠅⠘⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠟⢿⠉⠙⡄⠀⠈⠻⠟⠖⠛⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⢿⡄⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠈⠀⢸⡄⠀⠀⠑⠂⠠⡄⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⠀⠀⡄⢸⡇⠀⠀⡇
⠀⠀⢀⣤⣀⡴⠖⠀⡞⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢸⣰⠞⠙⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢸⡀⠀⠀⢿
⠀⠀⢍⢻⡏⠀⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⢸⡿⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⢈⣸⣏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣷⠀⢀⣰⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢿⠀⠀⠀⣿
⠀⠀⠘⢿⡏⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡁⣾⠿⣷⡾⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⡀⡏
⠀⠀⠀⢸⣡⠂⢾⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⣿⣿⡟⣷⡾⣿⠿⠾⠃⠀⠀⠀⠀⠀⠀⠀⢀⠘⡇⠀⢀⡇
⠀⠀⡇⢸⡏⢀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠋⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⡇
⠀⠀⣧⠀⣇⣼⣽⡇⠀⢀⡀⠀⠀⠀⣼⠀⠀⠀⠀⣸⣿⣿⣾⣿⡅⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠘⡇
⠀⢠⣿⢠⣿⣼⣿⠣⣀⠔⡴⠀⠀⠀⠆⡀⠀⠀⠀⢻⣿⣟⣿⢻⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⢂
⠀⠀⢹⣻⡇⠻⣿⣷⣏⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠛⠛⢡⠸⠙⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄⠀⠘⡄
⠀⠀⢸⢸⡇⠈⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡎⢦⢃⡻⠀⠀⠤⠤⠀⠀⠀⣤⠀⠀⠀⠀⢠⠀⠈⠀⠀⠃
⠀⠀⠀⠀⣹⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢈⣇⣾⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢇
⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡛⢺⣿⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⣄⠂⢸
⠀⠀⠀⠀⢹⡇⢰⡀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣿⡟⢻⣿⡀⠐⠂⠰⠉⠀⠀⠀⠀⠀⠀⠀⣿⣥⡀⢸⣧⠈⣼
⠀⠀⠀⠀⢀⣿⠀⡇⠀⢰⡇⠀⠀⠀⠈⡆⠀⠀⠀⠀⠸⡿⠃⠈⠻⠆⠀⢈⡵⠀⠀⠀⠀⠀⠀⠀⠀⡟⣼⡇⢰⡿⠀⡏
⠀⠀⠀⠀⢸⡏⢰⠃⠀⠘⣇⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠐⠀⢀⢻
⠀⠐⠀⠀⠈⡇⠘⡄⠀⠀⢹⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠞⠁⠀⠀⠀⠠⠀⠸⣿⡇⠀⠀⢿⣿⢲⡄
⠀⢀⠀⠀⢰⠇⠀⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢘⣿⣜⣡
⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢋⡵⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠈⠀⠀⢀⠈⣿⣿⣋⠄
⣿⣿⠿⠿⡿⢿⠿⡿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠐⠂⠤⢤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠲⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠀⠈⠳⠙⠲⡔⠒⠛⠛⠛⠓⠲⠶⠒⠂⠈⠉⠉⠁⠀⠀⠀⠠⣤⣀⡀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡀⠀⠀⡷
⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢒⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢰⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⢰⠀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠐⠮⣄⣀⣀⣀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠙⢾⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣾⠀⣉⠀⠀⣀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⡏⠛⣿⣧⡀⠈⠂⠶⠶⣦⣌⡉⠉⠉⠘⠛⠉⢻⠀⣼⣟⠷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠣⣼⣿⣿⣿⣶⣤⠶⡚⠅⠀⠉⠓⠀⠀⠀⠀⣻⡿⠿⢿⡤⢰⡀⠉⠁
⠀⠀⠀⠀⠀⠀⣰⣣⡴⠀⠀⠀⠀⠀⡰⠁⠀⠠⠖⠛⠛⠓⢦⣄⡛⢶⣤⠄⣉⣀⡀⠉⠀⠀⣠⠖⠓⠢⢄⢀⠀⠠⠣⣄⢠⠚⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⢁⡞⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠁⠐⠋⠐⠒⠀⠼⡆⠴⠤⡀⠀⠀⠀⢀⡜⠁⠀⠈⠑⠀⣁
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢢⡀⠀⣠⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⢠⡤⠀⠀⠀⠀⠀⠀⢀⠀⠉⣤⡄
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⡉⠂⢿⡀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣠⠂⢸
⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠿⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣄⡀⠀⠀⠀⠀⣀⣠⠜⠀⢸
⠀⠀⠀⠀⠀⢸⢘⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣰⠃⢰⣿⣿⣿⣿⣷⣦⣀⣭⠙⣿⠻⠋⠙⠿⠛⠯⠉⢯⠭⠭⠓⢿⡟⠙⠋⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⢿⠀⢸⣿⡟⠋⠛⢿⣿⣿⣿⣿⣿⠂⠀⣰⠷⣄⠤⠤⠤⠤⣆⠴⡟⠁⠀⠀⢀⡾⢦
⠀⠀⠀⠀⠀⠀⠀⢧⢠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢸⣿⢧⡀⠀⠀⠉⠙⢿⣿⣯⣴⣿⠁⠀⠀⡤⠀⠀⠀⠀⣠⠃⢀⣄⡠⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠈⠳⣄⠀⠀⠀⠀⠀⠀⠙⡆⠘⢻⡄⠱⠾⡤⠲⢀⠀⣿⣿⣿⣿⣀⣀⠢⠀⠀⠀⡠⠀⠄⠐⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠠⢄⠀⢹⡄⠈⢷⣤⣄⣀⣰⣯⠤⠄⠴⠤⠦⣤⡏⣠⠀⢠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠺⣅⠀⠀⠉⠉⠉⠀⠤⠄⠀⠀⠀⠀⠀⠈⠁⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠞⣧⠚⡧⠊⠑⠸⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠻⢿⣿⠿⠻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢯⣾⣎⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⣴⣿⣤⣤⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣿⡿⢿⣯⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣯⣤⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣳⣿⡟⢀⣾⣿⣿⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡼⠿⢿⠿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣵⣿⡟⠀⠾⢻⣿⣿⣷⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣀⠛⠛⠞⠛⠋⣠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣽⣿⣿⣷⣶⢀⣾⣿⣿⣿⣿⡽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠻⣿⣶⣤⣾⡿⠛⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣾⣿⣿⣿⣿⢁⣽⣿⣿⣿⣿⣿⣿⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⡿⠛⠛⠻⣿⣶⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠁⠀⠀⠀⠀⠀⠙⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⣿⡁⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣀⣠⣼⣿⣷⣄⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢃⣴⣾⣿⣿⣿⠟⠳⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣹⣿⣿⠟⣛⡛⢿⣿⣿⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣵⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣷⡹⣿⡇⣾⣿⣿⡆⣿⡿⣰⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⢠⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠹⢷⣬⣙⣩⣼⠿⣱⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣙⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⠃⠀⠉⠛⠋⠁⠀⠻⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣶⣴⠶⠤⠤⣤⣄⡀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠋⢀⡀⣹⣆⠰⡶⠶⢶⡿⢿⣿⣿⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⣿⡧⠸⣿⣿⡿⢃⣠⠾⠿⠛⣷⣶⣍⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠋⢰⣟⠁⢠⡞⠋⣡⡾⣶⣵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⣿⡿⠙⣿⡿⠛⢉⣤⡿⢿⣟⠿⠷⡦⠙⢰⣿⣿⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣁⠀⠹⣏⠀⣾⠋⣠⣿⡛⣿⣍⡽⣦⣀⣿⣿⣯⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢟⣿⣿⠟⠀⣴⡟⠁⢸⠇⣰⠿⡟⢧⣙⣻⣷⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⣿⡏⠀⣼⣿⡿⠁⠈⢠⣟⠀⠹⣆⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⠁⢹⣿⣷⡏⠀⠘⠛⠛⠛⣿⣶⠋⠩⠭⣉⠛⠻⣯⣝⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⡀⡈⠟⣿⣇⣶⢀⣠⠄⠀⠹⣿⣦⡀⠵⠒⡨⠀⢈⡿⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠘⣿⣷⣷⡀⠙⣿⣿⣿⣿⣴⣦⣤⠌⠻⣿⣿⣾⠿⠿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣏⠛⢦⣌⠛⢿⣿⣿⡟⠁⠀⠀⢠⣀⣀⣀⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠓⢄⠙⢿⡇⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣷⡄⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣤⣷⣣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣯⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢤⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣿⣿⣼⠟⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣙⣻⠻⣷⣦⣤⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⡷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣿⡿⢻⣿⣹⣿⠸⣎⠱⠏⢽⣏⠠⠘⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣿⢾⡀⠀⠀⠀⠀⠀⣀⣤⠶⢾⣿⣛⡿⠖⣛⣻⣌⣶⣍⠺⡶⢦⣰⣼⣿⣿⣀⣀⠠⢍⣿⡇⠘⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⣠⢶⡟⡹⢃⣘⣩⣯⠀⠻⣉⣹⣿⣶⣿⢿⣿⣿⡿⠛⠛⠙⠛⠙⠛⠻⠿⢴⣦⣾⠟⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣾⣤⣴⡾⢛⣓⡈⡷⢁⡿⢛⣁⣰⣶⠞⢿⣿⣿⠿⣿⣥⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣯⡿⠛⠄
⠀⠀⠀⠀⠀⣀⡤⠖⣔⡲⣦⢤⣤⣾⡍⢛⡩⠥⠺⠎⢁⣠⡴⠎⣀⠟⠵⣎⣹⣆⣶⣾⠟⣾⣵⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣾
⠀⠀⢀⡴⢎⠱⠒⣀⠆⠭⢸⡏⣿⡷⣻⡄⠴⠚⠚⣋⢈⡅⠤⠶⣤⣜⣿⣿⡿⠞⠉⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦
⢀⠴⠋⠀⠄⠂⢁⠠⢹⠀⢪⣇⣽⡯⣽⣶⣷⣿⣿⣿⣷⣶⣶⡯⠿⠟⠋⠁⠀⠀⠀⠉⠻⣿⣷⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉
⣏⡠⠀⠠⠠⠍⠭⠭⠌⣐⡢⣶⡿⠿⣿⣿⣯⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣾⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣤
⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠟⠋⠀⠀⠈⠙⢿⣿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⣻⣿⣿⣿⣝⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢹⣿⠞⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡷⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣛⣽⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠶⠦⠶⠞⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡛⡿⢿⣣⡀